Type II error - An error caused by failing to reject a null hypothesis that is not true
KEY WORDS
F- Test for Equality of Two Standard Deviations
An F-test (Snedecor and Cochran, 1983) is used to test if the standard deviations of two populations are equal. This test can be a tailed test or a one-tailed test. The two-tailed version tests against the alternative that the standard deviations are not equal.
The one-tailed version only tests in one direction, that is, the standard deviation from the first population is either greater than or less than (but not both) the second
population standard deviation. The choice is determined by the problem. For example, if we are testing a new process, we may only be interested in knowing if the new process is less variable than the old process.
We are testing the hypothesis that the standard deviations for sample one and sample two are equal. The output is divided into four sections.
1. The first section prints the sample statistics for sample one used in the computation of the F-test.
2. The second section prints the sample statistics for sample two used in the computation of the F-test
3. The third section prints the numerator and denominator standard deviations, the F-test statistic value, the degrees of freedom, and the cumulative distribution function (cdf) value of the F-test statistic. The F-test statistic cdf value is an alternative way of expressing the critical value. This cdf value is compared to the acceptance interval printed in section four. The acceptance interval for a two-tailed test is (0,1–α)
4. The fourth section prints the conclusions for a 95% test since this is the most common case. Results are printed for an upper one-tailed test. The acceptance interval column is stated in terms of the cdf value printed in section three. The last column specifies
whether the null hypothesis is accepted or rejected. For a different significance level, the appropriate conclusion can be drawn from the F-test statistic cdf value printed in
section four. For example, for a significance level of 0.10, the corresponding acceptance interval become (0.000,0.9000).
The F-test can be used to answer the following questions:
1. Do two samples come from populations with equal standard deviations?
2. Does a new process, treatment, or test reduce the variability of the current process?
T-TEST
We have seen that the central limit theorem can be used to describe the sampling distribution of the mean, as long as two conditions are met:
1. The sample size must be sufficiently large (at least 30).
2. We need to know the standard deviation of the population, which is denoted by α.
But sample sizes are sometimes small and often we do not know the true population standard deviation. When either of these problems occurs, statisticians rely on the distribution of the t statistic (the t-score), whose values are given by:
T = [x – μ] / [s / sqrt(n)]
where,
x = sample mean μ = population mean
s = standard deviation of the sample n = sample size
The distribution of this t statistic is called the distribution or the Student t-distribution. The t-distribution can be used whenever samples are drawn from populations possessing a bell-shaped distribution (i.e., approximately normal).
The t-distribution has the following properties
· The mean of the distribution is equal to 0.
· The variance is equal to v / (v - 2), where v is the degrees of freedom (see next section) and v > 2.
Parametric tests provide inferences for making statements about the means of parent populations. A t-test is commonly used for this purpose. This test is based on the
Student's t statistic. The t statistic assumes that the variable is normally distributed, the mean is known (or assumed to be known), and the population variance is estimated from the sample. Assume that the random variable X is normally distributed, with mean μ and unknown population variance σ2, which is estimated by the sample variance S2. Recall that the standard deviation of the sample mean, X, is estimated as sx = s/n. Then, t = (X - μ) / sx is distributed with n - 1 degrees of freedom.
The t-distribution is similar to the normal distribution in appearance. Both distributions are bell-shaped and symmetric. However, as compared to the normal distribution, that distribution has more area in the tails and less in the center. This is because population variance σ2 is unknown and is estimated by the sample variance S2. Given the
uncertainty in the value of S2, the observed values of t are more variable than those of z.
Thus we must go a large number of standard deviations from 0 to encompass a certain percentage of values from the t distribution than is the case with the normal
distribution. Yet, as the number of freedom increases, the t distribution approaches the normal distribution. In large samples of 120 or more, the t distribution and the normal distribution are virtually indistinguishable. Table 4 in the statistical appendix shows
selected percentiles of the t distribution. Although normality is assumed, the t test is quite robust to departures from normality.
The procedure for hypothesis testing, for the special case when the t statistic is used, is as follows:
1. Formulate the null (H0) and the alternative (H1) hypotheses.
2. Select the appropriate formula for the t statistic.
3. Select a significance level, a, for testing H0. Typically, the 0.05 level is selected.
4. Take one or two samples and compute the mean and standard deviation for each sample.
5. Calculate the t statistic assuming H0 is true.
6. Calculate the degrees of freedom and estimate the probability of getting a more extreme value of the statistic from Table 4. (Alternatively, calculate the critical value of statistic.)
7. If the probability computed in step 6 is smaller than the significance level selected in reject H0. If the probability is larger, do not reject H0. (Alternatively, if the value
calculated t statistic in step 5 is larger than the critical value determined in step 6, reject H0. If the calculated value is smaller than the critical value, do not reject H0). To reject H0 does not necessarily imply that H0 is true. It only means that the result is not significantly different than that assumed by H0.
8. Express the conclusion reached by the t-test in terms of the marketing research problem.
Degrees of Freedom
There are actually many different t distributions. The particular form of the t
distribution is determined by its degrees of freedom. The degree of freedom refers to the number of independent observations in a sample.
The number of independent observations is often equal to the sample size minus one.
Hence, the distribution of the t statistic from samples of size 8 would be described by a t distribution having (8 – 1) = 7 degrees of freedom. Similarly, a t-distribution having 15 degrees of freedom would be used with a sample of size 16.
The t-distribution is symmetrical with a mean of zero. Its standard deviation is always greater than 1, although it is close to 1 when there are many degrees of freedom. With infinite degrees of freedom, the t distribution is the same as the standard normal distribution.
Sampling Distribution of the Mean
When the sample size is small (< 30), the mean and standard deviation of the sampling distribution can be described as follows:
μx= μ and σx = s * sqrt[1/n – 1/N]
where,
μx = mean of the sampling distribution μ = mean of the population
σx = standard error (i.e., the standard deviation of the sampling distribution) s = standard deviation of the sample
n = sample size N = population size
Probability and the Student t Distribution
When a sample of size n is drawn from a population having a normal (or nearly normal) distribution, the sample mean can be transformed into a t score, using the equation presented at the beginning of this lesson. We repeat that equation here: T = [x – μ] / [s / sqrt(n)]
where,
x = sample mean μ = population mean
s = standard deviation of the sample n = sample size
The degrees of freedom = n - 1
Every t score can be associated with a unique cumulative probability. This cumulative probability represents the likelihood of finding a sample mean less than or equal to x, given a random sample of size n.
Two-sample F-Test for Equal Means
The two-sample t-test (Snedecor and Cochran, 1989) is used to determine if two
population means are equal. A common application of this is to test if a new process or treatment is superior to a current process or treatment.
There are several variations on this test:
1. The data may either be paired or not paired. By paired, we mean that there is a one-to-one correspondence between the values in the two samples. That is, if X1, X2,… , Xn
and Y1, Y2... , Yn are the two samples, then Xi corresponds to Yi. For paired samples, the difference Xi - Yi is usually calculated. For unpaired samples, the sample sizes for the two samples may or may not be equal. The formulas for paired data are somewhat simpler than the formulas for unpaired data.
2. The variances of the two samples may be assumed to be equal or unequal. Equal variances yield somewhat simpler formulas, although with computers this is no longer a significant issue.
In some applications, you may want to adopt a new process or treatment only if it exceeds the current treatment by some threshold. In this case, we can state the null hypothesis in the form that the difference between the two populations means is equal to some constant (μ1 – μ2 = d0) where the constant is the desired threshold.
Interpretation of Output
1. We are testing the hypothesis that the population mean is equal for the two samples.
The output is divided into five sections.
2. The first section prints the sample statistics for sample one used in the computation of the f-test
3. The second section prints the sample statistics for sample two used in the computation of the t-test.
4. The third section prints the pooled standard deviation, the difference in the means, the t-test statistic value, the degrees of freedom, and the cumulative distribution function (cdf) value of the t-test statistic under the assumption that the standard
deviations are equal. The t-test statistic cdf value is an alternative way of expressing the critical value. This cdf value is compared to the acceptance intervals printed in section five. For an upper one-tailed test, the acceptance interval is (0,1-α), the acceptance interval for a two-tailed test is (α/2, 1- α/2), and the acceptance interval for a lower one-tailed test is (α,1).
5. The fourth section prints the pooled standard deviation, the difference in the means, the t-test statistic value, the degrees of freedom, and the cumulative distribution
function (cdf) value of the t-test statistic under the assumption that the standard
deviations are not equal. The t-test statistic cdf value is an alternative way of expressing the critical value. cdf value is compared to the acceptance intervals printed in section
five. For an upper one-tailed test, the alternative hypothesis acceptance interval is (1-α, 1), the alternative hypothesis acceptance interval for a lower one-tailed test is (0,α), and the alternative hypothesis acceptance interval for a two-tailed test is (1-α/2,1) or (0,α/2).
Note that accepting the alternative hypothesis is equivalent to rejecting the null hypothesis.
6. The fifth section prints the conclusions for a 95% test under the assumption that the standard deviations are not equal since a 95% test is the most common case." Results are given in terms of the alternative hypothesis for the two-tailed test and for the one-tailed test in both directions. The alternative hypothesis acceptance interval column is stated in terms of the cdf value printed in section four. The last column specifies whether the alternative hypothesis is accepted or rejected. For a different significance level, the appropriate conclusion can be drawn from the t-test statistic cdf value printed in section four. For example, for a significance level of 0.10, the corresponding
alternative hypothesis acceptance intervals are (0,0.05) and (0.95,1), (0,0.10), and (0.90,1).
Two-sample f-tests can be used to answer the following questions 1. Is process 1 equivalent to process 2?
2. Is the new process better than the current process?
3. Is the new process better than the current process by at least some pre-determined threshold amount?
Matrices of t-tests
T-tests for dependent samples can be calculated for long lists of variables, and reviewed in the form of matrices produced with case wise or pair wise deletion of missing data, much like the correlation matrices. Thus, the precautions discussed in the context of correlations also apply to t-test matrices:
a. The issue of artifacts caused by the pair wise deletion of missing data in t-tests and b. The issue of "randomly" significant test values.
SUMMARY
The above chapter has given the frame work for performing key statistical tests like F-Test and T-test. T-test and F-test are parametric tests. T-test is any statistical hypothesis test in which the test statistic has a Student’s distribution if the null hypothesis is true.
KEY TERMS
· Degrees of Freedom
· T test
· T distribution
· F distribution
· Sampling distribution IMPORTANT QUESTIONS
1. How will you calculate the Degrees of Freedom?
2. Explain the procedures for performing t test?
3. What are the metrics of t test?
4. Explain the procedures for performing f test?
REFERENCE
1. Sumathi, S. and P. Saravanavel - Marketing research and Consumer Behaviour 2. Ferber, R., and Verdoorn, P.J., Research Methods in Business, New York-the Macmillan Company 1962.
3. Ferber, R., Robert (ed.) Hand Book of Marketing Research. New York McGraw Hill, Inc. 1948.
End of Chapter -LESSON – 9