Statistics: Overview
Statistics: Univariate Inferential Tests
Requirements: two normally distributed but independent populations, σ is unknown
Hypothesis test
Formula:
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where x 1, and x 2 are the means of the two samples, Δ is the hypothesized difference between the population means (0 if testing for equal means), s1 and s2 are the standard deviations of the two samples, and n1 and n2are the sizes of the two samples. The number of degrees of freedom for the problem is the smaller of n1− 1 and n2– 1.
Example 1 (one-tailed test): An experiment is conducted to determine whether intensive tutoring (covering a great deal of material in a fixed amount of time) is more effective than paced tutoring (covering less material in the same amount of time). Two randomly chosen groups are tutored separately and then administered proficiency tests. Use a significance level of α < .05.
null hypothesis: H0: μ1 ≤ μ2
or H0: μ1 − μ2 ≤ 0
alternative hypothesis: Ha: μ1 ≥ μ2
or: Ha: μ1 − μ2 ≥ 0
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The degrees of freedom parameter is the smaller of (12 – 1) and (10 – 1), or 9. Because this is a one-tailed test, the alpha level (.05) is not divided by two. The next step is to look up t.05,9 in the t-table, which gives a critical value of 1.833. The computed t of 1.166 does not exceed the tabled value, so the null hypothesis cannot be rejected. This test has not provided statistically significant evidence that intensive tutoring is superior to paced tutoring.
Confidence interval for comparing two means
Formula:
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where a and b are the limits of the confidence interval, x 1 and x 2 are the means of the two samples, tα/2,df is the value from the t-table corresponding to half of the desired alpha level, s1 and s2 are the standard deviations of the two samples, and n1 and n2 are the sizes of the two samples. The degrees of freedom parameter for looking up the t-value is the smaller of n1 – 1 and n2 – 1.
Example 2: Estimate a 90 percent confidence interval for the difference between the number of raisins per box in two brands of breakfast cereal.
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The difference between x 1 and x 2 is 102.1 – 93.6 = 8.5. The degrees of freedom is the smaller of (6 – 1) and (9 – 1), or 5. A 90 percent confidence interval is equivalent to an alpha level of .10, which is then halved to give .05. According to the t-table, the critical value for t.05,5 is 2.015. The interval may now be computed.
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You can be 90 percent certain that Brand A cereal has between 2.81 fewer and 19.81 more raisins per box than Brand B. The fact that the interval contains 0 means that if you had performed a test of the hypothesis that the two population means are different (using the same significance level), you would not have been able to reject the null hypothesis of no difference.
Pooled variance method
If the two population distributions can be assumed to have the same variance—and therefore the same standard deviation—
s1 and
s2 can be pooled together, each weighted by the number of cases in each sample. Although using pooled variance in a
t-test is generally more likely to yield significant results than using separate variances, it is often hard to know whether the variances of the two populations are equal. For this reason, the pooled variance method should be used with caution. The formula for the pooled estimator of σ2 is
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where s1 and s2 are the standard deviations of the two samples and n1 and n2 are the sizes of the two samples.
The formula for comparing the means of two populations using pooled variance is
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where x 1 and x 2 are the means of the two samples, Δ is the hypothesized difference between the population means (0 if testing for equal means), sp2 is the pooled variance, and n1 and n2 are the sizes of the two samples. The number of degrees of freedom for the problem is
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Example 3 (two-tailed test): Does right- or left-handedness affect how fast people type? Random samples of students from a typing class are given a typing speed test (words per minute), and the results are compared. Significance level for the test: .10. Because you are looking for a difference between the groups in either direction (right-handed faster than left, or vice versa), this is a two-tailed test.
null hypothesis: H0: μ1 = μ2
or: H0: μ1 – μ2 = 0
alternative hypothesis: Ha: μ1 ≠ μ2
or: Ha: μ1 – μ2 ≠ 0
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First, calculate the pooled variance:
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Next, calculate the
t-value:
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The degrees-of-freedom parameter is 16 + 9 – 2, or 23. This test is a two-tailed one, so you divide the alpha level (.10) by two. Next, you look up t.05,23 in the t-table, which gives a critical value of 1.714. This value is larger than the absolute value of the computed t of –1.598, so the null hypothesis of equal population means cannot be rejected. There is no evidence that right- or left-handedness has any effect on typing speed.
