10. The t-test for two independent samples - One-tailed example using tables

Most engaged couples expect or at least hope that they will have high levels of marital satisfaction. However, because 54% of first marriages end in divorce, social scientists have begun investigating influences on marital satisfaction. [Data source: This data was obtained from the National Center for Health Statistics.]

Suppose a social psychologist sets out to look at the role of race in relationship longevity. She decides to measure marital satisfaction in a group of African American couples and a group of Latino couples. She chooses the Marital Satisfaction Inventory, because it refers to "partner" and "relationship" rather than "spouse" and "marriage," which makes it useful for research with both traditional and nontraditional couples. Higher scores on the Marital Satisfaction Inventory indicate greater satisfaction. There is one score per couple. Assume that these scores are normally distributed and that the variances of the scores are the same among African American couples as among Latino couples.

The psychologist thinks that African American couples will have less relationship satisfaction than Latino couples. She identifies the null and alternative hypotheses as:

H: AfricanAmericancouplesAfricanAmericancouples LatinocouplesLatinocouples

H: AfricanAmericancouplesAfricanAmericancouples< LatinocouplesLatinocouples

This is aone- tailed test.

The psychologist collects the data. A group of 25 African American couples scored an average of 26.3 with a sample standard deviation of 8 on the Marital Satisfaction Inventory. A group of 23 Latino couples scored an average of 20.3 with a sample standard deviation of 9. Use the t distribution table. To use the table, you will first need to calculate the degrees of freedom.

The degrees of freedom are46 .

The t distribution

	Proportion in One Tail
	0.25	0.10	0.05	0.025	0.01	0.005
	Proportion in Two Tails Combined
df	0.50	0.20	0.10	0.05	0.02	0.01
1	1.000	3.078	6.314	12.706	31.821	63.657
2	0.816	1.886	2.920	4.303	6.965	9.925
3	0.765	1.638	2.353	3.182	4.541	5.841
4	0.741	1.533	2.132	2.776	3.747	4.604
5	0.727	1.476	2.015	2.571	3.365	4.032
6	0.718	1.440	1.943	2.447	3.143	3.707
7	0.711	1.415	1.895	2.365	2.998	3.499
8	0.706	1.397	1.860	2.306	2.896	3.355
9	0.703	1.383	1.833	2.262	2.821	3.250
10	0.700	1.372	1.812	2.228	2.764	3.169
11	0.697	1.363	1.796	2.201	2.718	3.106
12	0.695	1.356	1.782	2.179	2.681	3.055
13	0.694	1.350	1.771	2.160	2.650	3.012
14	0.692	1.345	1.761	2.145	2.624	2.977
15	0.691	1.341	1.753	2.131	2.602	2.947
16	0.690	1.337	1.746	2.120	2.583	2.921
17	0.689	1.333	1.740	2.110	2.567	2.898
18	0.688	1.330	1.734	2.101	2.552	2.878
19	0.688	1.328	1.729	2.093	2.539	2.861
20	0.687	1.325	1.725	2.086	2.528	2.845
21	0.686	1.323	1.721	2.080	2.518	2.831
22	0.686	1.321	1.717	2.074	2.508	2.819
23	0.685	1.319	1.714	2.069	2.500	2.807
24	0.685	1.318	1.711	2.064	2.492	2.797
25	0.684	1.316	1.708	2.060	2.485	2.787
26	0.684	1.315	1.706	2.056	2.479	2.779
27	0.684	1.314	1.703	2.052	2.473	2.771
28	0.683	1.313	1.701	2.048	2.467	2.763
29	0.683	1.311	1.699	2.045	2.462	2.756
30	0.683	1.310	1.697	2.042	2.457	2.750
40	0.681	1.303	1.684	2.021	2.423	2.704
60	0.679	1.296	1.671	2.000	2.390	2.660
120	0.677	1.289	1.658	1.980	2.358	2.617
	0.674	1.282	1.645	1.960	2.326	2.576
	0.50	0.20	0.10	0.05	0.02	0.01

With = 0.01, the critical t-score (the value for a t-score that separates the tail from the main body of the distribution, forming the critical region) is-2.423 . (Note:If your df value is not included in this table, look up the critical values for both surrounding df values and select the larger t value to use as your critical t-score. If you fail to reject the null hypothesis, you can later check the smaller t value to decide whether to interpolate. However, for the purposes of this problem, you can just assume that if your t statistic is not more extreme than the larger t value, you will not reject the null hypothesis. Also, the table includes only positive t values. Since the t distribution is symmetrical, for a one-tailed test where the alternative hypothesis is less than, simply negate the t value provided in the table.)

To calculate the t statistic, you first need to calculate the estimated standard error of the difference in means. To calculate this estimated standard error, you first need to calculate the pooled variance. The pooled variance is . The estimated standard error of the difference in means is . (Hint:For the most precise results, retain four decimal places from your calculation of the pooled variance to calculate the standard error.)

Calculate the t statistic. The t statistic is . (Hint:For the most precise results, retain four decimal places from your previous calculation to calculate the t statistic.)

The t statistic lie in the critical region for a one-tailed hypothesis test. Therefore, the null hypothesis is . The psychologist conclude that African American couples have less relationship satisfaction than Latino couples.