3. Hypothesis testing with sample means (small samples)

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

Suppose a sociologist sets out to study the role of race in relationship longevity. She measures relationship longevity in a random sample of African American couples and in a random sample of Latino couples and compares the data. Assume that relationship longevity is normally distributed and that the variance in longevity is approximately the same among African American couples as among Latino couples.

Step 1: Do the data meet the test requirements?

A. Is there independent random sampling?

No

Yes

B. What is the level of measurement of the variables?

Nominal

Ordinal

Interval-ratio

C. Is the sampling distribution normally distributed?

Yes

No

Step 2: State the null hypothesis.

The sociologist thinks that African American couples will have less relationship longevity than Latino couples. Identify the null and research hypotheses:

H0

:
H1

:

This is a -tailed test.

Step 3: Select the sampling distribution and establish the critical region.

The sociologist collects data from one sample of African American couples and another of Latino couples. With small samples, the t distribution is used to establish the critical region because the combined less than 100.

Relationship Longevity

African American Couples	Latino Couples
X1

= 18.4 years

X2

= 24.1 years

s1

= 8 years

s2

= 9 years

N1

= 25

N2

= 23

Use the t distribution table that follows to find 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. To use the table, you will first need to calculate the degrees of freedom (df). The degrees of freedom are . With = 0.01, the critical t-score is .

df	Proportion in One Tail	0.25	0.10	0.05	0.025	0.01	0.005
Proportion in Two Tails	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

Step 4: Computing the test statistic.

To calculate the t statistic, you first need to estimate the population variance. You can estimate the population variance by calculating a weighted sample variance (s(XX)

). The pooled estimate of the standard deviation is . The value of the test statistic is t = . (Hint:For the most precise results, retain four decimal places from your previous calculation to calculate the t statistic.)

Step 5: Making a decision and interpreting the results of the test.

The t statistic in the critical region for this hypothesis test. Therefore, the sociologist the null hypothesis. The sociologist conclude that relationships are shorter for African American couples than for Latino couples.