Given two dependent random samples with the following results:
| Population 1 | 22 | 39 | 20 | 34 | 43 | 17 | 45 |
|---|---|---|---|---|---|---|---|
| Population 2 | 33 | 45 | 35 | 46 | 35 | 27 | 37 |
Use this data to find the $98 \%$ confidence interval for the true difference between the population means. Assume that both populations are normally distributed.
Step 1 of 4 : Find the point estimate for the population mean of the paired differences. Let $x_1$ be the value from Population 1 and $x_2$ be the value from Population 2 and use the formula $d=x_2-x_1$ to calculate the paired differences. Round your answer to one decimal place.
Answer (2)
The point estimate for the population mean of the paired differences is approximately 5.4. This is calculated from the paired differences obtained by subtracting the values of Population 1 from Population 2. The mean of these differences is then rounded to one decimal place.
;
Calculate the paired differences between the two populations.
Find the mean of the paired differences: d ˉ ≈ 5.4 .
Determine the critical t-value for a 98% confidence interval with 6 degrees of freedom: t 0.01 = 3.143 .
Calculate the 98% confidence interval: ( − 5.95 , 16.75 ) .
Explanation
Problem Analysis We are given two dependent random samples and asked to find the 98% confidence interval for the true difference between the population means. We will calculate the paired differences, the mean of the differences, the standard deviation of the differences, the critical t-value, and finally the confidence interval.
Calculate Paired Differences First, we calculate the paired differences d = x 2 − x 1 :
d 1 = 33 − 22 = 11 d 2 = 45 − 39 = 6 d 3 = 35 − 20 = 15 d 4 = 46 − 34 = 12 d 5 = 35 − 43 = − 8 d 6 = 27 − 17 = 10 d 7 = 37 − 45 = − 8
So the paired differences are: 11, 6, 15, 12, -8, 10, -8.
Calculate the Mean of Differences Next, we calculate the sample mean of the paired differences, d ˉ .
d ˉ = 7 11 + 6 + 15 + 12 − 8 + 10 − 8 = 7 38 ≈ 5.4
Calculate Standard Deviation of Differences Now, we calculate the sample standard deviation of the paired differences, s d .
First, we calculated that the standard deviation of the differences is approximately 9.55.
Find the Critical t-Value We need to find the critical value t α /2 for a 98% confidence interval with n − 1 = 7 − 1 = 6 degrees of freedom. Since the confidence level is 98%, α = 1 − 0.98 = 0.02 , so α /2 = 0.01 . We need to find t 0.01 with 6 degrees of freedom.
We found that t 0.01 = 3.143 .
Calculate the Margin of Error Now we calculate the margin of error E using the formula E = t α /2 ∗ ( s d / n ) .
E = 3.143 ∗ 7 9.554 ≈ 3.143 ∗ 2.646 9.554 ≈ 3.143 ∗ 3.610 ≈ 11.35
Calculate Confidence Interval Next, we calculate the lower and upper bounds of the confidence interval:
Lower bound: d ˉ − E = 5.4 − 11.35 = − 5.95 Upper bound: d ˉ + E = 5.4 + 11.35 = 16.75
Therefore, the 98% confidence interval for the true difference between the population means is approximately (-5.95, 16.75).
Final Answer The 98% confidence interval for the true difference between the population means is approximately (-5.95, 16.75).
Examples
Imagine you're testing a new drug to lower blood pressure. You measure patients' blood pressure before and after taking the drug. The paired difference is the 'after' minus 'before' blood pressure for each patient. A confidence interval for the mean difference tells you the range within which the true average change in blood pressure likely falls, helping you determine if the drug is effective.
