Deze samenvatting is gebaseerd op het studiejaar 2013-2014.
I.1 Concepts of estimation
The objective of estimation is to determine the approximate value of a population parameter on the basis of a sample statistic. For instance, the sample mean is used to estimate the population mean. The sample mean is referred to as the estimator of the population mean. Once the sample mean has been computed, its value is called the estimate.
Sample data can be used to estimate a population parameter in two ways:
A point estimator draws inferences about a population by estimating the value of an unknown parameter using a single value or point. There are three drawbacks to using this estimator:
it is virtually certain that the estimate will be wrong. (The probability that a continuous random variable will equal a specific value is 0.)
generally, researchers need to know how close the estimator is to the parameter.
point estimators do not have the capacity to reflect the effects of larger sample sizes.
An interval estimator draws inferences about a population by estimating the value of an unknown parameter using an interval. The interval estimator is affected by the sample size.
An unbiased estimator of a population parameter is an estimator whose expected value is equal to that parameter. In other words, if a researcher was to take an infinite number of samples and calculate the value of the estimator in each sample, the average value of the estimator would equal the parameter. Thus, on average, the sample statistic is equal to the parameter.
The sample mean figuur 1 is an unbiased estimator of the population mean µ. It was stated before that .Figuur 2 The sample proportion is an unbiased estimator of the population proportion because Figuur 3. The difference between two sample means is an unbiased estimator of the difference between two population means because .
Figuur 4 Previously the sample was variance as:
Figuur 5
The reason for choosing to divide by n – 1 is to make E (s2)= σ2 that this definition makes the sample variance an unbiased estimator of the population variance.
If the sample variance was defined using n in the denominator, the resulting statistic would be a biased estimator of the population variance, one whose expected value is less than the parameter. If an estimator is unbiased researchers can be sure that its expected value equals the parameter; however, it does not say how close the estimator is to the parameter. Another desirable quality is consistency - an unbiased estimator is said to be consistent if the difference between the estimator and the parameter grows smaller as the sample size grows larger.
The measure which is used to estimate closeness is the variance (or the standard deviation). Thus, Figuur 1 is a consistent estimator of μ, because the variance Figuur 1 of is σ2 / n This implies that as n grows larger, the variance of Figuur 1 grows smaller. As a consequence, an increasing proportion of sample means falls close to μ .
Similarly, Figuur 6 is a consistent estimator of P because it is unbiased and the variance of Figuur 6 is, P (1- p) / n which grows smaller as n grows larger.
The last desirable quality is relative efficiency, which compares two unbiased estimators of a parameter: If there are two unbiased estimators of a parameter, the one whose variance is smaller is said to be relatively more efficient.
I.2 Estimate population mean
Confidence Interval Estimator of µ
Figuur 7
The probability 1 – a is called the confidence level.
Figuur 8 is called the lower confidence limit (LCL).
Figuur 9 is called the upper confidence limit (UCL).
The confidence interval estimator is often represented as:
Figuur 10
where the minus sign defines the lower confidence limit and the plus sign defines the upper confidence limit.
To apply this formula a researcher has to specify the confidence level, 1 – a from which he or she can determine a, a/2, za/2 (from Table 3 in Appendix B). Because the confidence level is the probability that the interval includes the actual value of μ , researchers usually set 1 – a close to 1 (i.e. between .90 and .99)
The four most commonly used confidence levels and their associated value of zα/2 are listed:
1 – α zα/2
.90 z.05 = 1.645
.95 z.025 = 1.96
.98 z.01 = 2.33
.99 z.005 = 2.575
Let’s assume that the confidence level is 1 – α = .98. Consequently, α = .02, α/2 = .01, and zα/2 = z.01 = 2.33. The resulting confidence interval estimator is called the 98% confidence interval estimator of µ.
The confidence interval estimate of µ cannot be treated as a probability statement about µ. However, the confidence interval estimator is a probability statement about the sample mean. It states that there is 1 – α probability that the sample mean will equal to a value such that interval Figuur 1 will include the population mean. Once the sample mean is computed, the interval acts as the lower and upper limits of the interval estimate of the population mean.
The width of the confidence interval estimate is a function of the population standard deviation, the confidence level, and the sample size. Thus:
Doubling the population standard deviation has the effect of doubling the width of the confidence interval estimate.
Decreasing the confidence level narrows the interval; increasing it widens the interval. Note: a large confidence level is generally desirable since that means a larger proportion of confidence interval estimates that will be correct in the long run. As a general rule 95% confidence is considered “standard.”
Increasing the sample size fourfold decreases the width of the interval by half. A larger sample size provides more potential information. The increased amount of information is reflected in a narrower interval.
I.3 Selecting sample size
Sampling error is the difference between the sample and the population that exists only because of the observations that happened to be selected for the sample. The sampling error can also be defined as the difference between an estimator and a parameter. This difference can also be defined as the error of estimation. This can be express as the difference between Figuur 1 and µ.
The error of estimation is less than Zα/2 σ/√n. It means that Zα/2 σ/√n is the maximum error of estimation that a researcher is willing to tolerate. This value is labeled B, which stands for the bound on the error of estimation. That is,
B = Z a/2 ( σ) / ( √n )
The equation for n can be solved if the population standard deviation σ, the confidence level 1 – α, and the bound on the error of estimation B are known.
Thus,
Sample size to Estimate a Mean n = ( z a/2 σ/ B) 2
J.1 Concepts of hypothesis testing
Hypothesis testing is the second general procedure of making inferences about a population
The most important concepts in hypothesis testing are the following:
There are two hypotheses. One is called the null hypothesis and is represented by H0 and the other is called the alternative or research hypothesis and is represented by H1. The null hypothesis will always state that the parameter equals the value specified in the alternative hypothesis
The testing procedure begins with the assumption that the null hypothesis is true.
The goal of the process is to determine whether there is enough evidence to infer that the alternative hypothesis is true.
There are two possible decisions:
There are two possible errors. A Type I error occurs when a true null hypothesis is rejected. A Type II error is defined as not rejecting a false null hypothesis. The probability of a Type I error is denoted by α, which is also called the significance level. The probability of a Type II error is denoted by β. The error probabilities α and β are inversely related, meaning that any attempt to reduce one will increase the other.
P(Type I error) = α
P(Type II error) = β
The hypotheses are often set up to reflect a manager’s decision problem while the null hypothesis represents status quo. The next step in the process is to randomly sample the population and calculate the sample mean. This is called the test statistic. The tests statistics is the criterion upon which researchers base decision about the hypotheses. The test statistic is based on the best estimator of the parameter (i.e. the best estimator of a population mean is the sample mean).
If the test statistic’s value is inconsistent with the null hypothesis, researchers reject the null hypothesis and infer that the alternative hypothesis is true. In the absence of sufficient evidence, the null hypothesis cannot be rejected in favour of the alternative.
It seems reasonable to reject the null hypothesis in favour of the alternative if the value of the sample mean is large relative to the population mean. If, for instance, the sample mean was calculated to be 300, while the population mean was expected to be 100, it would be quite apparent that the null hypothesis is false and has to be rejected it.
On the other hand, values of Figuu 1 close to 100, for instance 97, do not allow researchers to reject the null hypothesis because it is entirely possible to observe a sample mean of 97 from a population whose mean is 100. To make a decision about this sample mean, they have to set up the rejection region.
J.2 Testing population mean
The rejection region is a range of values such that if the test statistics falls into that range, a researcher has to reject the null hypothesis in favour of the alternative hypothesis.
To calculate the rejection region, let’s assume that the value of the sample mean is just large enough to reject the null hypothesis as Figuur 11. The rejection region is:
Figuur 12
Since a type I error is defined as rejecting a true null hypothesis, and the probability of committing a Type I error is a, it follows that
Figuur 13
The sampling distribution of Figuur 1 is normal or approximately normal, with mean μ and standard deviation σ / √n As a result, can Figuur 1 be standardized to obtain the following probability:
Figuur 14
Z a is the value of a standard normal random variable such that
p (Z > za) = a
Since both probability statements involve the same distribution (standard normal) and the same probability (a), it follows that the limits are identical. Thus,
Figuur 15
To calculate the rejection region, Figuur 11 a researcher has to have values of sample size (n), standard deviation (σ), population mean (µ), and a value of (a) , the significance level.
Let’s assume that the sample mean was computed to be 110 and the rejection region was calculated to be 105.47. Because the test statistic (sample mean) is in the rejection region (it is greater than 105.47), the null hypothesis has to be rejected.
The preceding test used the test statistic ;Figuur 1 as a result, the rejection region had to be set up in terms of Figuur 1 . An easier method specifies that the test statistic be the standardized value of Figuur 1. That is, researchers can use the standardized test statistic:
figuur 16
and the rejection region consists of all values of that are Z greater than Za . Algebraically the rejection region is
Z > Z a
The standardized test statistic can be used, thus
Z > Z a = z 05 = 1.645 (when the significance level is 5%)
Consequently, the value of the test statistic has to be calculated (using the formula ) Figuur 17 and the result should be compared to the rejection region (1.645 in this case). If the result is greater than 1.645, a researcher has to reject the null hypothesis.
The conclusions drawn from using the test statistic Figuur 1 and the standardized test statistic Z are identical.
When a null hypothesis is rejected, the test is said to be statistically significant at whatever significance level the test was conducted. For instance, it can be said that the test was significant at the 5% significance level.
The p-value of a test is the probability of observing a test statistic at least as extreme as the one computed given that the null hypothesis is true. An example of calculating p-value can be seen below.
Figuur 18
In this case the probability of observing a sample mean at least as large as 178 from a population whose mean is 170 is .0069, which is very small. In other words, it is an unlikely event. If the hypothesis is H0 = 170, such p-value gives a reason to reject the null hypothesis and support the alternative.
The p-value of a test provides valuable information because it is a measure of the amount of statistical evidence that supports the alternative hypothesis.
How small does the p-value have to be to infer that the alternative hypothesis is true, depends on a number of factors, including the costs of making Type I and Type II errors. If the cost of Type I error is high, researchers attempt to minimize its probability. In the rejection region method, they do so by setting the significance level quite low (e.g. 1%). Using the p-value method, researchers need the p-value to be quite small, providing sufficient evidence to reject the null hypothesis.
p-values can be translated using the following descriptive terms:
If the p-value is less than .01, researchers say that there is overwhelming evidence to infer that alternative hypothesis is true. It can also be said that the test is highly significant.
If the p-value lies between .01 and .05, there is strong evidence to infer that the alternative hypothesis is true. The result is deemed to be significant.
If the p-value is between .05 and .10, researchers say that there is weak evidence to indicate that the alternative hypothesis is true. When the p-value is greater than 5%, it is said that the result is not statistically significant.
When the p-value exceeds .10, it is said that there is no evidence to infer that the alternative hypothesis is true.
The p-value can be used to make the same type of decisions made in the rejection region method. The rejection region method requires the decision maker to select a significance level from which the rejection region is constructed. He then decides to reject or not reject the null hypothesis. Another way of making that type of decision is to compare the p-value with the selected value of the significance level. If the p-value is less than α, the p-value is judged to be small enough to reject the null hypothesis. If the p-value is greater than α, the null hypothesis cannot be rejected.
Conclusion of a Test of Hypothesis
If a researcher rejects the null hypothesis, he or she concludes that there is enough statistical evidence to infer that the alternative hypothesis is true.
If a researcher does not reject the null hypothesis, he or she concludes that there is not enough statistical evidence to infer that the alternative hypothesis is true.
In one-tail tests, the rejection region is located only one tail of the sampling distribution. The p-value is also computed by finding the area in one tail of the sampling distribution.
A two-tail test is conducted whenever the alternative hypothesis specifies that the mean is not equal to the value stated in the null hypothesis, that is, when the hypotheses assume the following form:
H0 : μ = μ 0
H1 : μ = μ 0
There are two one-tail tests. Researchers conduct a one-tail test that focuses on the right tail of the sampling distribution whenever they want to know whether there is enough evidence to infer that the mean is greater than the quantity specified by the null hypothesis, that is, when the hypotheses are:
H0 : μ = μ 0
H1 : μ > μ 0
The second one-tail test involves the left tail of the sampling distribution. It is used when the researcher wants to determine whether there is enough evidence to infer that the mean is less than the value of the mean stated in the null hypothesis. The resulting hypotheses appear in this form:
H0 : μ = μ 0
H1 : μ < μ 0
The test statistic and the confidence interval estimator are both derived from the sampling distribution. The confidence interval estimator figuur 19 can be used to test hypotheses.
This process is equivalent to the rejection region approach. However, instead of finding the critical values of the rejection region and determining whether the test statistic falls into the rejection region, researchers compute the interval estimate and determine whether the hypothesized value of the mean falls in to the interval.
The test of hypothesis is based on the sampling distribution of the sample statistic. The result of a test of hypothesis is a probability statement about the sample statistic. It is assumed that the population mean is specified by the null hypothesis. Researchers then compute the test statistic and determine how likely it is to observe this large (or small) a value when the null hypothesis is true. If the probability is small, it can be concluded that the assumption that the null hypothesis is true is unfounded and it is rejected.
When researchers calculate the value of the test statistic, they are also measuring the difference between the sample statistic Figuur 1 and the hypothesized value of the parameter μ in terms of the standard error σ) / √n For instance, if the value of the test statistic was z = 1.19, this would mean that the sample mean is 1.19 standard errors above the hypothesized value of . μ The standard normal probability table shows that this value is not considered unlikely. As a result the null hypothesis should not be rejected.
J.3 Probability of type II error
A Type II error occurs when a false null hypothesis is not rejected. if figuur 1 is less than rejection region, the null hypothesis is not rejected. The probability of a Type II error is defined as:
Figuur 20 [value of the rejection region], given that the null hypothesis is false)
The condition that the null hypothesis is false, means only that the mean is not equal to μ. If a researcher wants to compute , β he needs to specify a value for μ . To calculate the probability of a Type II error, researchers have to express the rejection region in terms of the unstandardized test statistic , Figuur 1 and they have to specify a value for μ other than the one shown in the null hypothesis.
Effect of Changing a on β
By decreasing the significance level from 5% to 1%, a researcher will shift the critical value of the rejection region to the right and thus enlarge the area where the null hypothesis is not rejected. The probability of a Type II error increases. This illustrates the inverse relationship between the probabilities of Type I and Type II errors. If a researcher wants to decrease the probability of a Type I error (by specifying a small value of a), he will increase the probability of a Type II error. In applications where the cost of a Type I error is considerably larger than the cost of a Type II error, this is appropriate.
In fact, a significance level of 1% or less is probably justified. However, when the cost of a Type II error is relatively large, a significance level of 5% or more may be appropriate.
A statistical test of hypothesis is effectively defined by the significance level and the sample size (both selected by the researcher). A researcher can judge how well the test functions by calculating the probability of a Type II error at some value of the parameter.
If he believes that the cost of a Type II error is high and thus that the probability is too large, he has two ways to reduce the probability. He can either increase the value (a) of (however, this would result in an increase in the chance of making a Type I error) or increase the sample size. By increasing the sample size, a researcher reduces the probability of a Type II error. By reducing the probability of a Type II error, he makes this type of error less frequently. For this reason, larger sample sizes allow researchers to make better decisions in the long run.
Another way of expressing how well a test performs is to report its power: the probability of its leading a researcher to reject the null hypothesis when it is false. Thus, the power of a test is 1- β.
When more than one test can be performed in a given situation, it is preferred to use the test that is correct more frequently. If (given the same alternative hypothesis, sample size, and significance level) one test has a higher power than a second test, the first test is said to be more powerful.
J.4 The road ahead
In the chapters that follow, different statistical techniques are presented employed by statistics practioners. The real challenge of the subject lies in being able to define the problem and identify which statistical method is the most appropriate one to use. Every statistic method has some specific objective (5 addressed in this book: describe a population, compare two populations, compare 2+ populations, analyze the relationship between two variables and analyze the relationship among 2+ variables).
K.1 Inference about a population mean
If the population mean is unknown, so is the population standard deviation. Consequently, the previous sampling distribution cannot be used. As an alternative, researchers substitute the sample standard deviation (S) in place of the unknown population standard deviation (S). The result is called a t-statistic. It has been shown showed that the t-statistic defined as
Figuur 21
is Student (t) distributed when the sampled population is normal.
Test statistic for μ when (s) is unknown
When the population standard deviation is unknown and the population is normal, the test statistic for testing hypotheses about μ is
Figuur 22
which is Student t distributed with v = n – 1 degrees of freedom
Confidence interval estimator of μ when σ is unknown
Figuur 23 v = n – 1
The t-statistic is Student t distributed if the population from which a researcher had sampled is normal. However, statisticians have shown that if the population is nonnormal, the results of the t-test and confidence interval estimate are still valid provided that the population is not extremely nonnormal. To check this requirement, a researcher has to draw the histogram and determine whether it is far from bell shaped.
When the population is small, the test statistic and interval estimator have to be adjusted using the finite population correction factor. However, in populations that are large relative to the sample size, the correction factor can be ignored. Large populations are defined as populations that are at least 20 times the sample size.
Finite populations allow researchers to use the confidence interval estimator of a mean to produce a confidence interval estimator of the population total. To estimate the total, it is necessary to multiply the lower and upper confidence limits of the estimate of the mean by the population size. Thus, the confidence interval estimator of the total is:
Figuur 24
The Student t distribution is based on using the sample variance to estimate the unknown population variance. The sample variance is defined as
Figuur 25
To compute S2 , a researcher must first determine .Figuur 1 Sampling distributions are derived by repeated sampling from the same population.
To repeatedly take samples to compute S2 , he or she can choose any numbers for the first n - 1 observations in the sample. However, he or she has no choice on the th n value because the sample mean must be calculated first. Let’s assume N – 3 that and Figuur 26 has to be found.X1 and X2 can assume any values without restriction. However, X3 must be such that Figuur 26. For instance, if ..X1 = 5 and X2 = 11 then X3 must equal 14. Therefore, there are only 2 degrees of freedom in this selection of the sample. It is said that 1 degree of freedom was lost because Figuur 1 had to be calculated. Note: the denominator in the calculation of s2 is equal to the number of degrees in freedom.
The t -statistic like the z -statistic measures the difference between the sample me Figuur 1 an and the hypothesized value of μ in terms of the number of standard errors. However, when the population standard deviation α is unknown, the standard error is estimated by . s / √n
The t -statistic has two variables: the sample mean and Figuur 1 the sample standard deviation s, both of which will vary from sample to sample. Because of the greater uncertainty, the t -statistic will display greater variability.
K.2 Inference about a population variance
The estimator of σ 2 is the sample variance; that is, S2 is an unbiased, consistent estimator of σ 2. It has been shown that the sum of squared deviations from the mean Figuur 27 [which is equal to ] (n- 1) S2 divided by the population variance is chi-squared distributed with v = n – 1 degrees of freedom provided that the sampled population is normal. The statistic
X 2 = ( n-1) S2 /σ 2
is called the chi-squared statistic (C2 -statistic).
The formula that describes the sampling distribution is the formula of the test statistic. The test statistic used to test hypotheses about σ 2 is
X 2 = ( n-1) S2 / σ 2
Which is chi-squared distributed with v = n – 1 degrees of freedom when the population random variable is normally distributed with variance equal to σ 2.
Confidence interval estimator of σ 2
Lower confidence limit (LCL) = ( n-1) S2 / X2a/2
Upper confidence limit (UCL) = ( n-1) S2 / X12a/2
Like the t-test and estimator of μ , the chi-squared test and estimator of theoretic σ 2 ally require that the sample population be normal.
In practice, however, the technique is valid as long as the population is not extremely nonnormal. The extent of nonnormality can be determined by drawing the histogram.
K.3 Inference about a population proportion
The statistic used to estimate and test the population proportion is the sample proportion defined as
Figuur 28
Where Figuur 1 is the number of successes in the sample and is (n) the sample size. The sampling distribution of Figuur 29 is approximately normal with mean (P) and standard deviation √p( 1 – p) / n (provided that (np) and n (1- p)are greater than 5). This sampling distribution is expressed as
Figuur 30
The same formula also represents the test statistic.
Test statistic for (P)
Figuur 31
which is approximately normal for NP and n (1 - p) greater than 5
Confidence interval estimator (P)
Figuur 32
which is valid provided that Figuur 33 and Figuur 34 are greater than 5
To produce the confidence interval estimator of the total, a researcher has to multiply the lower and upper confidence limits of the interval estimator of the proportion of successes by the population size. The confidence interval estimator of the total number of successes in a large finite population is
Figuur 35
Sample size to Estimate a Proportion:
Figuur 36
To solve for , n a researcher has to know Figuur 29 . Unfortunately, this value is unknown, because the sample has not yet been taken. Two methods can be used to solve for n:
This, in turn, results in a conservative value of , and n as a result, the confidence interval will be no wider than the interval . If figuur 39, when the sample is drawn,figuur 29 does not equal .5, the confidence interval estimate will be better (that is, narrower) than planned. If it turns out that = .5, the interval estimate is . Figuur 39 If not, the interval estimate will be narrower. For instance, if it turns out that Figuur 40 , the estimate Figuur 41 is , which is better than planned.
If a researcher some idea about the value Figuur 29 of, he can use that quantity to determine n. For instance, if he believes that Figuur 29 will turn out to be approximately .3, he can solve for n . Note: this produces a smaller value n of (thus reducing sampling costs) than does the previous method. If Figuur 29 actually lies between .3 and .7, however, the estimate will not be as good as wanted, because the interval will be wider than desired.
Wilson Estimators
When applying the confidence interval estimator of a proportion when success is a relatively rare event, it is possible to find no successes, especially if the sample size is small. To illustrate, let’s assume a sample of 100 produced X = 0 which means that . Figuur 42 The 95% confidence interval estimator of the proportion of successes in the population becomes
Figuur 43
This implies that if a researcher does not find successes in the sample, then there is no chance of finding a success in the population. Drawing such a conclusion from virtually any sample size is unacceptable. The Wilson estimate denoted Figuur 44 (pronounced -til P de) is computed by adding 2 to the number of successes in the sample and 4 to the sample size.
Confidence interval estimator of P using the Wilson Estimate
Figuur 45
L.1 Inference about the difference between two means: independent samples
In order to test and estimate the difference between two population means, researchers draw random samples from each of two populations. Independent samples are defined as samples completely unrelated to one another. A researcher draws a sample size of size n1 from population 1 and a sample of size n2 from population 2. For each sample, he or she has to compute the sample means and sample variances.
The best estimator of the difference between two population means μ1 - μ2 is the difference between two sample means Figuur 46.
Sampling Distribution of Figuur 46
Figuur 46 is normally distributed if the populations are normal and approximately normal if the populations are nonnormal and the sample sizes are large.
The expected value of Figuur 46 is
Figuur 47
The variance of Figuur 46 is
Figuur48
The standard error of Figuur 46 is
√ ( σ 21 / n1) + (σ 22 / n2)
Thus,
Figuur 49
is a standard normal (or approximately normal) random variable. Consequently, the test statistic is
Figuur 49
And the interval estimator is
Figuur 50
However, these formulas are rarely used because the population variances σ 21 and σ22 are virtually always unknown. Consequently, it is necessary to estimate the standard error of the sampling distribution.
The way to do this depends on whether the two unknown population variances are equal. When they are equal, the test statistic is defined in the following way.
Tests Statistic for μ1 – μ2 when σ21 = σ22
Figuur 51 v = n1 = n2 – 2
where
s 2p = ( n1 – 1) s 21 + (n2 – 1) s 22 / n1 + n2 – 2
The quantity s 2p is called the pooled variance estimator. It is the weighted average of the two sample variances with the number of degrees of freedom in each sample used as weights. The requirement that the population variances be equal makes this calculation feasible, because only one estimate of the common value of S21 an S22 d is needed.
The test statistic is Student t distributed with n1 + n2 – 2 degrees of freedom, provided that the two populations are normal.
Confidence Interval Estimator of μ1 – μ2when σ 21 = σ 22
Figuur 52 v = n1 + n2 – 2
These formulas are referred to as the equal-variances test statistic and confidence interval estimator, respectively.
When the population variances are unequal, researchers cannot use the pooled variance estimate. Instead, they estimate each population variance with its sample variance.
Unfortunately, the sampling distribution of the resulting statistic
Figuur 53
is neither normally nor Student t distributed. However, it can be approximated by a Student t distribution with degree of freedom equal to
The test Figuur 54 atistic and confidence interval estimator are easily derived from the sampling distribution.
Test statistic for :ų1 ų2 when σ 21 = σ 22
Figuur 55 Figuur 56
Confidence interval estimator Ц1 Ц2 when σ 21 = σ 22
Figuur 57 Figuur 58
These formulas are referred to as the unequal-variances test statistic and confidence interval estimator, respectively.
Since σ 21 and σ 22 are unknown, a researcher cannot know for certain whether they are equal. However, he or she can perform a statistical test to determine whether there is evidence to infer that the population variances differ. A researcher has to conduct the F-test of the ratio of two variances.
Testing the population variances
The hypotheses to be tested are
H0 : σ 21 / σ 22 = 1
H1 : σ 21 / σ 22 σ 1
The test statistic is the ratio of the sample variances S21 / S22 which is F-distributed with degrees of freedom v1 = n1 – 1 and v2 = n2 – 2
The required condition for F-distribution is the same as that for the t-test of Ц1 - Ц2 , which is that both populations are normally distributed. This is a two-tail test so that the rejection region is
F > F a/2, v1,v2 or F > F 1-a/2, v1,v2
Thus, a researcher will reject the null hypothesis that states that the population variances are equal when the ratio of the sample variances is large or if it is small. Table 6 in Appendix B lists the critical values of the F distribution and defines “large” and “small.”
L.2 Observational and experimental data
A researcher can never have enough statistical evidence to conclude that the null hypothesis is true. This means that he or she can only determine whether there is enough evidence to infer that the population variances differ. Accordingly, a researcher adopts the following rule: he will use the equal-variances test statistic and confidence interval estimator unless there is evidence (based on the F-test of the population variances) to indicate that the population variances are unequal, in which case he will apply the unequal-variances test statistic and confidence interval estimator.
Both the equal-variances and unequal-variances techniques require that the populations are normally distributed. As before, a researcher can check to see whether the requirement is satisfied by drawing the histograms of the data.
When the normality requirement is unsatisfied, a researcher can use a nonparametric technique – the Wilcoxon rank sum test – to replace the equal-variances test of . Ц1 - Ц2 There is no alternative to the unequal-variances test of Ц1 - Ц2 when the populations are very nonnormal.
The value of the test statistic is the difference between the statistic Figuur 59 and the hypothesized value of the parameter Ц1 - Ц2 measured in terms of the standard error.
As was the case with the interval estimator of P, the standard error must be estimated from the data for all inferential procedures introduced. The method used to compute the standard error of Figuur 59 depends on whether the population variances are equal. When they are equal, researchers calculate and use the pooled variance estimator. S2p Thus, where possible, it is advantageous to pool sample data to estimate the standard error. S2p is a better estimator of the common variance than either S21 or S22 separately. When the two population variances are unequal, a researcher cannot pool the data and produce a common estimator – he must compute S21 and S22 and use them to estimate 21 and 22 respectively.
L.3 Inference about the difference between two means: matched pairs
An experiment may be designed in such a way that each observation in one sample is matched with an observation in the other sample. The matching is conducted by selecting, for instance, economics and marketing majors with similar GPAs. Thus, it is logical to compare the salary offers for them in each group. This type of experiment is called matched pairs. In such experimental design, the parameter of interest is the mean of the population of differences, which we label Цd.
Note: Цd. does in fact equal Ц1 - Ц2 , but researchers test Цd. because of the way the experiment is designed. Therefore, the hypotheses to be tested are:
H0 : Цd = 0
H1 : Цd > 0
Test statistic for Цd :
Figuur 60
which is Student t distributed with v = nD -1degrees of freedom, provided that the differences are normally distributed.
The confidence interval estimator of is Цd. derived using the usual form for the confidence interval.
Confidence interval estimator of Цd.
Figuur 61
The validity of the results of the t-test and estimator of Цd. depends on the normality of the differences (or large enough sample sizes). For instance, the histogram of the differences can be positively skewed, but not enough so that the normality requirement is violated.
If the differences are very nonnormal, t -test of Цd cannot be used. Researchers can, however, use a nonparametric technique – the Wilcoxon signed rank sum test for matched pairs.
Two most principles in statistics are:
The concept of analyzing sources of variation. For instance, by reducing the variation between salary offers in each sample, a researcher is able to detect a real difference between the two majors. This is an application of the more general procedure of analyzing data and attributing some fraction of the variation to several sources. A technique called the analysis of variance analyzes sources of variation in an attempt to detect real differences. In most applications of this procedure, researchers are interested in each source of variation and not simply in reducing one source. The process is referred to as explaining the variation.
Researchers can design data-gathering procedures in such a way that they can analyze sources of variation. The experiment can be organized so that the effects of those differences are mostly eliminated. It is also possible to design experiments that allow for easy detection of real differences and minimize the costs of data gathering.
Researchers make inferences about the ratio of two population variances because the sampling distribution is based on ratios rather than differences.
Two population variances are compared by determining the ratio; thus, the parameter is .21 and 22 The sample variance is an unbiased and consistent estimator of the population variance. The estimator of the parameter 21 and 22is the ratio of the two sample variances drawn from their respective populations . The S21 / S22 sampling distribution of 21 and 22s said to be F distributed provided that researchers have independently sampled from two normal populations. It has been shown that the ratio of two independent chi-squared variables divided by their degrees of freedom is F distributed.
The degrees of freedom of the F distribution are identical to the degrees of freedom for the two chi-squared distributions. (n-1) s2 / 2 chi-squared distributed, provided that the sampled population is normal.
M.1 Chi-squared goodness-of-fit test
A multinomial experiment is an extension of the binomial experiment, in which there are two or more possible outcomes per trial. A multinomial experiment is one possessing the following characteristics:
The experiment consists of a fixed number n of trials.
The outcome of each trial can be classified into one of k categories, called cells.
The probability pi that the outcome will fall into cell i remains constant for each trial. Moreover, .p1 + p2 + …... + p k = 1
Each trial of the experiment is independent of the other trials.
When k = 2, the multinomial experiment is identical to the binomial experiment. In a binomial experiment, researchers count the number of successes (which is labeled x) and failures. In a multinomial experiment, researchers count the number of outcomes falling into each of the k cells. Thus, they obtain a set of observed frequencies ƒ1,ƒ2,......, ƒk where ƒ1, is the observed frequency of outcomes falling into cell i , for . i = 1,2,....., k Because the experiment consists of n trials and an outcome must fall into some cell,
ƒ1,ƒ2,......, ƒk = n
Just as the number of successes x was used to draw inferences about p (by calculating the sample proportion Figuur 6, which is equal to x/n), the observed frequencies are used to draw inferences about the cell probabilities.
If the data are nominal and a researcher is interested in the proportions of all categories the experiment is recognized as a multinomial experiment, and the technique is identified as the chi-squared goodness-of-fit test. Because a researcher wants to know whether the values of each category changed, he needs to specify the initial values in the null hypothesis, e.g. H0 : P1 = 45, p2 = 40, p3 = 15
The alternative hypothesis states: H1: At least one Pi is not equal to its specified value.
In general, the expected frequency for each cell is given by
ei = npi
This expression is derived from the formula for the expected value of a binomial random variable.
If the expected frequencies ei and the observed frequencies a ƒ1 re quite different, it can be concluded that the null hypothesis is false, and is should be rejected. However, if the expected and observed frequencies are similar, the null hypothesis should not be rejected. The following test statistic measures the similarity of the expected and observed frequencies.
Chi-squared Goodness-of-fit Test:
Figuur 62
The sampling distribution of the test statistic is approximately chi-squared distributed with v = k – 1 degrees of freedom, provided that the sample size is large.
When the null hypothesis is true, the observed and expected frequencies should be similar, in which case the test statistic will be small. Thus, a small test statistic supports the null hypothesis. If the null hypothesis is untrue, some of the observed and expected frequencies will differ and the test statistic will be large. Consequently, the null hypothesis should be rejected when X2 is greater than X2a,k-1That is, the rejection region is
X2 - X2a,k-1
The actual sampling distribution of the test statistic defined previously is discrete, but it can be approximated by the chi-squared distribution provided that the sample size is large. This requirement is similar to the one imposed when the normal approximation to the binomial was used in the sampling distribution of a proportion. In that approximation np and n (1 -p)had to be 5 or more.
M.2 Chi-squared tests of a contingency table
A similar rule is imposed for the chi-squared test statistic. It is called the rule of five, which states that the sample size must be large enough so that the expected value for each cell must be 5 or more. Where necessary, cells should be combined to satisfy this condition.
The chi-squared test of a contingency table is used to determine whether there is enough evidence to infer that two nominal variables are related and to infer that difference exists between two or more populations of nominal variables.
The test statistic is the same as the one used to test proportions in the goodness-of-fit-test. That is, the test statistic is
Figuur 62
where K is the number of cells in the cross-classification table. Note: In the goodness-of-fit test, the null hypothesis lists values for the probabilities . Pi The null hypothesis for the chi-squared test of a contingency table states only that the two variables are independent. However, the probabilities are needed to compute the expected values ei , which in turn are needed to calculate the value of the test statistic. The probabilities must come from the data after it is assumed that the null hypothesis is true.
The expected frequency of the cell in row I and column j is
ei = Rowitotal * Column j total / Samplesize
In order to determine the rejection region, a researcher must know the number of degrees of freedom associated with the chi-squared statistic. The number of degrees of freedom for a contingency table with r rows and c columns is v= (r-1) (c-1)
Regression analysis is used to predict the value of one variable on the basis of other variables. The technique involves developing a mathematical equation or model that describes the relationship between the variable to be forecast, which is called the dependent variable, and variables that the researcher believes are related to the dependent variable. The dependent variable is labeled as Y, while the related variables are called independent variables and are labeled X1, X2,.....Xk here K the number of independent variables).X1, X2,.....Xk
N.1 Model
Some of the mathematical models related to the statistical concepts are:
Deterministic models, which allow researchers to determine the value of the dependent variable (on the left side of the equation) from the values of the independent variables.
What must be included in most practical models is a method to represent the randomness that is part of a real-life process. Such a model is called a probabilistic model.
The first-order linear model (or the simple linear regression model) is a straight-line model with one independent variable.
First-order linear model
y = β0 + β1x + E
where
y = dependent variable
X = independent variable
β0 = y-intercept
β1 = slope of the line (defined as rise/run)
E = error variable
The problem objective addressed by the model is to analyze the relationship between two variables, X nd ,Y of which must be interval. To define the relationship between Xnd Y researchers need to know the value of the coefficients β0 and β1 wever, these coefficients are population parameters, which are almost always unknown.
The parameters β0 and β1 are estimated in a way similar to the methods used to estimate the other parameters discussed so far. A researcher has to draw a random sample from the population of interest and calculate the sample statistics needed. However, because β0 and β1 represent the coefficients of a straight line, their estimators are based on drawing a straight line through the sample data. The straight line which is used to estimate β0 and β1 is the “best” straight line, best in the sense that it comes closest to the sample data points.
N.2 Estimating the coefficients
This best straight line, called the least squares line, is derived from calculus and is represented by the following equation:
Figuur 63
where b0 the yintercept, b1 the slope, and figuur 64 ondicted or fitted value of y.
The least squares method produces a straight line that minimizes the sum of the squared differences between the points and the line. The coefficients b0 and b1 are calculated so that the sum of squared deviations
Figuur 65
is minimized. That is, the values of Figuur 64 come closest to the observed values of .
y
Least squares line coefficients
b1 = Sxy / S 2 x
Figuur 66
where
Figuur 67
Figuur 68
Figuur 69
Figuur 70
A shortcut method to manually calculate the slope coefficient b1 is:
b1 = Sxy / S 2 x
Figuur 71
Figuur 72
It has been shown that b0 and b1 are unbiased estimators of b0 and b1 respectively.
O.1 Wilcoxon rank sum test
The Wilcoxon rank sum test is used to for example determine whether observations from 2 populations allow the researcher to conclude that the location of population 1 is to the left of the location of population 2. The Wilcoxon rank sum has the following characteristics:
The problem objective is to compare two populations.
The data is either ordinal or interval where the normality requirement necessary to perform the equal-variances t-test of Ц1 - Ц2 s unsatisfied.
The samples are independent.
Suppose that we have 2 samples and we want to test the following hypotheses:
H0: The two population locations are the same
H1: The location of population 1 is to the left of the location of population 2
The first step is to rank all the observations (in case of 2 samples and 3 observations each: rank 1 to the smallest observation and rank 6 to the largest observation). The second step is to calculate the sum of the ranks of each sample. The rank sum of sample 1 is denoted as T1, and the rank sum of sample 2 is denoted as T2. T1 is arbitrarily selected as the test statistics and labelled as T (T1=T). A small value of T indicated that most of the smaller observations are in sample 1 and that most of the larger observations are in sample 2. This would imply that the location of population 1 is to the left of the location of population 2.
If the null hypothesis is true and the two populations are identical, then it follows that each possible ranking is equally likely. We are trying to determine whether the value of the test statistic is small enough to reject the null hypothesis at the 5% significance level. Statisticians have generated the sampling distribution of T for various combinations of sample sizes. The critical values are provided in Table 9 in Appendix B.
When the sample size are larger then 10, the test statistic is approximately normally distributed with mean E(T) and standard deviation T where
E ( T) = nl ( nl + n2 + 1) / 2
And
σT = √ nl (n2(nl+ n2 +1) / 12
bijlage_1.pdf
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