Topic 6 Homework (Nonadaptive)
Question 3 of 11 (t point) I Question Attempt: 6 of Unlimited
Katherine
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A certain counselor wants to compare mean IQ scores for two different social groups. A random sample of 14 IQ scores from group 1 showed a mean of 119 and a standard deviation of 13 , while an independently chosen random sample of $10 \mathrm{IQ}$ scores from group 2 showed a mean of 81 and a standard deviation of 15 . Assuming that the populations of IQ scores are normally distributed for each of the groups and that the variances of these populations are equal, construct a $90 \%$ confidence interval for the difference $\mu_{1}-\mu_{2}$ between the mean $\mu_{1}$ of IQ scores of group 1 and the mean $\mu_{2}$ of IQ scores of group 2 . Then find the lower limit and upper limit of the $90 \%$ confidence interval.
Carry your intermediate computations to at least three decimal places. Round your responses to at least two decimal places. (If necessary, consult a list of formulas.)
Lower limit: $\square$
Upper limit: $\square$
Answer
The lower limit of the 90% confidence interval for the difference between the mean IQ scores of group 1 and group 2 is approximately 27.90, and the upper limit is approximately 48.10. Therefore, we are 90% confident that the true difference between the mean IQ scores of the two groups is between 27.90 and 48.10. So, the final answer is \(\boxed{27.90}\) and \(\boxed{48.10}\).
Step 1 ：Given values are: mean of group 1 (x1) = 119, standard deviation of group 1 (s1) = 13, sample size of group 1 (n1) = 14, mean of group 2 (x2) = 81, standard deviation of group 2 (s2) = 15, sample size of group 2 (n2) = 10.
Step 2 ：Calculate the t-score for a 90% confidence interval with 22 degrees of freedom. The t-score is approximately 1.717.
Step 3 ：Calculate the standard error (se) using the formula \(\sqrt{\frac{s1^2}{n1} + \frac{s2^2}{n2}}\). The standard error is approximately 5.880.
Step 4 ：Calculate the lower limit of the confidence interval using the formula \((x1 - x2) - t_{score} \times se\). The lower limit is approximately 27.90.
Step 5 ：Calculate the upper limit of the confidence interval using the formula \((x1 - x2) + t_{score} \times se\). The upper limit is approximately 48.10.
Step 6 ：The lower limit of the 90% confidence interval for the difference between the mean IQ scores of group 1 and group 2 is approximately 27.90, and the upper limit is approximately 48.10. Therefore, we are 90% confident that the true difference between the mean IQ scores of the two groups is between 27.90 and 48.10. So, the final answer is \(\boxed{27.90}\) and \(\boxed{48.10}\).
