9. The t test for two independent samples - One-tailed example using tables Most engaged couples expect or at least hope

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9. The t test for two independent samples - One-tailed exampleusing tables Most engaged couples expect or at least hope that theywill have high levels of marital satisfaction. However, because 54%of first marriages end in divorce, social scientists have beguninvestigating influences on marital satisfaction. [Data source:This data was obtained from the National Center for HealthStatistics.] Suppose a clinical psychologist sets out to look atthe role of age of entering the marriage in relationship longevity.She decides to measure marital satisfaction in a group of couplesmarried after age 30 and a group of couples married before age 30.She chooses the Marital Satisfaction Inventory, because it refersto “partner” and “relationship” rather than “spouse” and“marriage,” which makes it useful for research with bothtraditional and nontraditional couples. Higher scores on theMarital Satisfaction Inventory indicate greater satisfaction. Thereis one score per couple. Assume that these scores are normallydistributed and that the variances of the scores are the same amongcouples married after age 30 as among couples married before age30. The psychologist thinks that couples married after age 30 willhave greater relationship satisfaction than couples married beforeage 30. She identifies the null and alternative hypotheses as: H₀:μcouples married after age 30 couples married after age 30 μcouplesmarried before age 30 couples married before age 30 H₁: μcouplesmarried after age 30 couples married after age 30 μcouples marriedbefore age 30 couples married before age 30 This is a tailed test.The psychologist collects the data. A group of 51 couples marriedafter age 30 scored an average of 45.8 with a sample standarddeviation of 9 on the Marital Satisfaction Inventory. A group of 54couples married before age 30 scored an average of 40.6 with asample standard deviation of 8. Use the t distribution table. Touse the table, you will first need to calculate the degrees offreedom. The degrees of freedom are . The t distribution Proportionin One Tail 0.25 0.10 0.05 0.025 0.01 0.005 Proportion in Two TailsCombined df 0.50 0.20 0.10 0.05 0.02 0.01 1 1.000 3.078 6.31412.706 31.821 63.657 2 0.816 1.886 2.920 4.303 6.965 9.925 3 0.7651.638 2.353 3.182 4.541 5.841 4 0.741 1.533 2.132 2.776 3.747 4.6045 0.727 1.476 2.015 2.571 3.365 4.032 6 0.718 1.440 1.943 2.4473.143 3.707 7 0.711 1.415 1.895 2.365 2.998 3.499 8 0.706 1.3971.860 2.306 2.896 3.355 9 0.703 1.383 1.833 2.262 2.821 3.250 100.700 1.372 1.812 2.228 2.764 3.169 11 0.697 1.363 1.796 2.2012.718 3.106 12 0.695 1.356 1.782 2.179 2.681 3.055 13 0.694 1.3501.771 2.160 2.650 3.012 14 0.692 1.345 1.761 2.145 2.624 2.977 150.691 1.341 1.753 2.131 2.602 2.947 16 0.690 1.337 1.746 2.1202.583 2.921 17 0.689 1.333 1.740 2.110 2.567 2.898 18 0.688 1.3301.734 2.101 2.552 2.878 19 0.688 1.328 1.729 2.093 2.539 2.861 200.687 1.325 1.725 2.086 2.528 2.845 21 0.686 1.323 1.721 2.0802.518 2.831 22 0.686 1.321 1.717 2.074 2.508 2.819 23 0.685 1.3191.714 2.069 2.500 2.807 24 0.685 1.318 1.711 2.064 2.492 2.797 250.684 1.316 1.708 2.060 2.485 2.787 26 0.684 1.315 1.706 2.0562.479 2.779 27 0.684 1.314 1.703 2.052 2.473 2.771 28 0.683 1.3131.701 2.048 2.467 2.763 29 0.683 1.311 1.699 2.045 2.462 2.756 300.683 1.310 1.697 2.042 2.457 2.750 40 0.681 1.303 1.684 2.0212.423 2.704 60 0.679 1.296 1.671 2.000 2.390 2.660 120 0.677 1.2891.658 1.980 2.358 2.617 ∞ 0.674 1.282 1.645 1.960 2.326 2.576 0.500.20 0.10 0.05 0.02 0.01 With α = 0.01, the critical t-score (thevalue for a t-score that separates the tail from the main body ofthe distribution, forming the critical region) is . (Note: If yourdf value is not included in this table, look up the critical valuesfor both surrounding df values and select the larger t value to useas your critical t-score. If you fail to reject the nullhypothesis, you can later check the smaller t value to decidewhether to interpolate. However, for the purposes of this problem,you can just assume that if your t statistic is not more extremethan the larger t value, you will not reject the null hypothesis.Also, the table includes only positive t values. Since the tdistribution is symmetrical, for a one-tailed test where thealternative hypothesis is less than, simply negate the t valueprovided in the table.) To calculate the t statistic, you firstneed to calculate the estimated standard error of the difference inmeans. To calculate this estimated standard error, you first needto calculate the pooled variance. The pooled variance is . Theestimated standard error of the difference in means is . (Hint: Forthe most precise results, retain four decimal places from yourcalculation of the pooled variance to calculate the standarderror.) Calculate the t statistic. The t statistic is . (Hint: Forthe most precise results, retain four decimal places from yourprevious calculation to calculate the t statistic.) The t statisticlie in the critical region for a one-tailed hypothesis test.Therefore, the null hypothesis is . The psychologist conclude thatcouples married after age 30 have greater relationship satisfactionthan couples married before age 30.