Business Statistics : Chapter 11; Two-Sample Test of Hypothesis

Comparting Two Population Variances

  • Apply the F-distribution to test a hypothesis that two population variances are equal.

    F-Distribution: use to test assumption or hypothesis that two population variances are equal, and also when comparing several population means at the same time.

    F-Distribution Characteristics:

  • There is a family of F-distributions. A specific member of the family is determined by two parameters, degrees of freedom in numerator and in denominator.

  • Is Continuous. Values of F can assume infinite number of values starting at 0.

  • Cannot be negative. Must be 0 or above.

  • Is positively skewed. Long tail of distribution if at right side. As degrees of freedom increase in both numerator and denominator, distribution becomes a normal distribution.

  • Is asymptotic. As values of F increase, distribution approaches x-axis but does not ever touch it.

    Testing a Hypothesis of Equal Population Variances: first application of F-distribution. Test hypothesis that variance of one normal population is equal to variance of another normal population.

        Step 1.) State null and alternate hypothesis. Null is that variances are equal, alternate is that the variances differ.

        Step 2.) Conduct test. Select random samples of observations from two populations.

            Test Statistic: F = Sample Variance of Population 1 / Sample Variance of Population 2

*Sample variances above are found by squaring the sample standard deviations*

    Critical Value of F: for a two-tailed test, found by dividing significance level in half (a/2) then finding appropriate degrees of freedom.

Two-Sample Tests of Hypothesis: Independent Samples

  • Test a hypothesis that two independent population means are equal, assuming that the population standard deviations are known.

    Variance of the Distribution of Differences in Means =

        (Standard Deviation of Population 1 Squared / Number of Samples in Population ) + (Standard Deviation of Population 2 Squared / Number of Samples in Population 2)

    Two-Sample Test of Means with Known Standard Deviation = find test statistic

    z= Mean of Population 1 - Mean of Population 2 / Square root of Variance of the Distribution of Differences in Means

    Assumptions necessary for using above formula: both follow normal distributions, both are unrelated and independent, both standard deviations of populations are known

    Steps:

        Step 1.) State null and alternate hypothesis.

        Step 2.) Select level of significance, which is the probability that the null hypothesis is rejected if it is actually true.

        Step 3.) Find test statistic. Use z-distribution as test statistic. (Formula above.)

        Step 4.) Find Decision Rule, which in this case is the z critical value. Found by using t-distribution table, under correct level of significance row and column for the correct alpha/degree of freedom. Reject if value exceeds this value.

        Step 5.) make Decision on Null Hypothesis. Does test stat (z value) exceed the decision rule?

        Step 6.) Interpret Results. Find P-value by finding corresponding value for z value with table (b.3 in textbook) by .5000.

Comparing Population Means with Estimated Population Standard Deviations

  • Test a hypothesis that two independent population means are equal, with estimated population standard deviations.

    Two-Sample Pooled Test: requires that population standard deviations are estimated with sample standard deviations. Three differences between this test and the previous Two-Sample Test of Means.

  • First, test hypothesis that population variances are equal

  • Next, if failure to reject hypothesis that population variances are equal, assume they are equal and average or “pool” the sample variances

  • Use t-distribution as test statistic

    Pooled Variance = (# observations in first sample)(variance of first sample) + (# observations in second sample)(variance of second sample) / # observations in first sample + # observations in second sample -2

* Sample variance is found by square of standard deviation of sample*

    Two Sample test of Means with Unknown Standard Deviations:

        t = first sample mean - second sample mean / square root of (pooled variance * (1/ # observations in first sample + 1/# observations in second sample))

    Unequal Population Standard Deviations: when test for equal population variances is done as the hypothesis is rejected.

    Text Statistic for no difference in means, unequal variances=

        t = mean of first sample - mean of second sample / square root of variance of sample 1/ # observations in sample 1 + variance of sample 2/ # observations in sample 2

    Degrees of Freedom for Unequal Variance Test:

        df = (sample 1 variance / sample 1 observations) + (sample 2 variance / sample 2 observations) /                 (sample 1 variance / sample 1 observations squared / sample 1 observations) -1 + (sample 2                 variance / sample 2 observations squared / sample 2 observations) -1

Two-Sample Test of Hypothesis: Dependent Samples

  • Test a hypothesis about the mean population difference between paired or dependent observations.

    Paired t-Test: t = mean of differences between paired observations / standard deviation of differences / square root of number of paired observations

Two-Sample Tests about Proportions

  • Test a hypothesis about two population proportions.

Two-Sample Test of Proportions:

            z = proportion in first sample - proportion in second sample / square root of pooled proportion (q- pooled proportion) / # observations in first sample + square root of pooled proportion (q- pooled proportion) / # observations in second sample

            Pooled Proportion = # possessing trait in first sample + # possessing trait in second sample / # observations in first sample + # observations in second sample

Excel Functions

Conducting an F Test: use data analysis, F-Test Two Sample for Variance and select rows of raw data. Checks labels and change alpha to level of significance.

Conducting Two-Sample t Test (independent populations, assuming equal variances):

  • When raw data available, use data analysis, T-Test two-sample assuming unequal variances. Set hypothesized mean difference to 0, and alpha to level of significance.

  • If there is no raw data, sample mean and standard deviation can be used instead. Use t.dist and t.inv.

Conducting Paired t-Test with Dependent Samples: use t-test paired two-sample for means.