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. Suppose a clinical psychologist sets out to look at the role of economic hardship in relationship longevity. She decides to measure marital satisfaction in a group of couples living above the poverty level and a group of couples living below the poverty level. 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 couples living above the poverty level as among couples living below the poverty level. The psychologist thinks that couples living above the poverty level will have greater relationship satisfaction than couples living below the poverty level. She identifies the null and alternative hypotheses as: H₀: μcouples living above the poverty levelcouples living above the poverty level μcouples living below the poverty levelcouples living below the poverty level H₁: μcouples living above the poverty levelcouples living above the poverty level μcouples living below the poverty levelcouples living below the poverty level This is a tailed test. The psychologist collects the data. A group of 39 couples living above the poverty level scored an average of 51.1 with a sample standard deviation of 9 on the Marital Satisfaction Inventory. A group of 31 couples living below the poverty level scored an average of 45.2 with a sample standard deviation of 12. Use the t distribution table. To use the table, you will first need to calculate the degrees of freedom. The degrees of freedom are . 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 α = 0.05, 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 . (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 . The t statistic lie in the critical region for a one-tailed hypothesis test. Therefore, the null hypothesis is . The psychologist conclude that couples living above the poverty level have greater relationship satisfaction than couples living below the poverty level.

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. Suppose a clinical psychologist sets out to look at the role of economic hardship in relationship longevity. She decides to measure marital satisfaction in a group of couples living above the poverty level and a group of couples living below the poverty level. 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 couples living above the poverty level as among couples living below the poverty level. The psychologist thinks that couples living above the poverty level will have greater relationship satisfaction than couples living below the poverty level. She identifies the null and alternative hypotheses as: H₀: μcouples living above the poverty levelcouples living above the poverty level μcouples living below the poverty levelcouples living below the poverty level H₁: μcouples living above the poverty levelcouples living above the poverty level μcouples living below the poverty levelcouples living below the poverty level This is a tailed test. The psychologist collects the data. A group of 39 couples living above the poverty level scored an average of 51.1 with a sample standard deviation of 9 on the Marital Satisfaction Inventory. A group of 31 couples living below the poverty level scored an average of 45.2 with a sample standard deviation of 12. Use the t distribution table. To use the table, you will first need to calculate the degrees of freedom. The degrees of freedom are . 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 α = 0.05, 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 . (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 . The t statistic lie in the critical region for a one-tailed hypothesis test. Therefore, the null hypothesis is . The psychologist conclude that couples living above the poverty level have greater relationship satisfaction than couples living below the poverty level.

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Suppose a clinical psychologist sets out to look at the role of economic hardship in relationship longevity. She decides to measure marital satisfaction in a group of couples living above the poverty level and a group of couples living below the poverty level. 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 couples living above the poverty level as among couples living below the poverty level.

The psychologist thinks that couples living above the poverty level will have greater relationship satisfaction than couples living below the poverty level. She identifies the null and alternative hypotheses as:

H₀: μcouples living above the poverty levelcouples living above the poverty level μcouples living below the poverty levelcouples living below the poverty level

H₁: μcouples living above the poverty levelcouples living above the poverty level μcouples living below the poverty levelcouples living below the poverty level

This is a tailed test.

The psychologist collects the data. A group of 39 couples living above the poverty level scored an average of 51.1 with a sample standard deviation of 9 on the Marital Satisfaction Inventory. A group of 31 couples living below the poverty level scored an average of 45.2 with a sample standard deviation of 12. Use the t distribution table. To use the table, you will first need to calculate the degrees of freedom.

The degrees of freedom are .

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 α = 0.05, 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 . (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 .

The t statistic lie in the critical region for a one-tailed hypothesis test. Therefore, the null hypothesis is . The psychologist conclude that couples living above the poverty level have greater relationship satisfaction than couples living below the poverty level.

Features Features Normal distribution is characterized by two parameters, mean (µ) and standard deviation (σ). When graphed, the mean represents the center of the bell curve and the graph is perfectly symmetric about the center. The mean, median, and mode are all equal for a normal distribution. The standard deviation measures the data's spread from the center. The higher the standard deviation, the more the data is spread out and the flatter the bell curve looks. Variance is another commonly used measure of the spread of the distribution and is equal to the square of the standard deviation.

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