Question

Given two independent random samples with the following results:

n₁=11 n₂=8
xˉ₁=174 xˉ₂=191
s₁=34 z₂=22

Use this data to find the 90 क्ष contidence interval for the true difference between the population means. Assume that the population variances are equal and that the two populations are normally distributed. Step 2 of 3 : Find the margh of error to be used in construxting the confidence interval, Rosind your answer to sic tiecimvilaces. Answer How to enter youranswer (ogerd in new window?

Answer 1

To construct a 90% confidence interval for the difference between two population means given two samples, the margin of error is calculated by finding the pooled variance, computing the standard error, and multiplying this by the critical t-value for 90% confidence.

Step-by-step explanation:

The question revolves around constructing a 90% confidence interval for the difference between two population means, given two independent samples. To find the margin of error for the confidence interval, we need to use the sample statistics provided and the t-distribution since the population variances are unknown but assumed equal.

We first calculate the pooled variance, using the formula:

1. Find the squared values of the provided standard deviations (s1^2 and s2^2).
2. Use the formula for pooled variance: Sp^2 = ((n1 - 1)*s1^2 + (n2 - 1)*s2^2) / (n1 + n2 - 2).
3. Calculate the square root of the pooled variance to get the pooled standard deviation (Sp).

Next, we compute the standard error of the mean difference by:

1. Applying the formula: SE = Sp * sqrt(1/n1 + 1/n2).
2. Find the critical t-value for a 90% confidence level with n1 + n2 - 2 degrees of freedom.
3. Multiply the t-value by the standard error to obtain the margin of error.

The confidence interval is then given by the sample mean difference plus or minus the margin of error: (xˉ1 - xˉ2) ± margin of error.

Note that the t-value required can be found in a t-distribution table or calculated using statistical software.