Question

Given two independent random samples with the following results:

n1=18x‾1=108s1=14
n2=10x‾2=86s2=18
Use this data to find the 90%
confidence interval for the true difference between the population means. Assume that the population variances are not equal and that the two populations are normally distributed.


Step 1 of 3 : Find the critical value that should be used in constructing the confidence interval. Round your answer to three decimal places.

Answer 1

The critical value to be used in constructing the confidence interval is approximately 1.711.

How to find the critical value

To find the critical value for constructing the confidence interval, use the t-distribution since the sample sizes are small and the population variances are assumed to be unequal.

The degrees of freedom for this case can be calculated using the formula:

`df = (s_1^2/n_1 + s_2^2/n_2)^2 / [(s_1^2/n_1)^2 / (n_1 - 1) + (s_2^2/n_2)^2 / (n_2 - 1)]`

Plugging in the given values:

n₁ = 18,
`\bar{x}`₁ = 108, s₁ = 14

n₂ = 10,
`\bar{x}`₂ = 86, s₂ = 18

We can calculate the degrees of freedom (df) as follows:

df =
`[(14^2/18 + 18^2/10)^2] / [(14^2/18)^2 / (18 - 1) + (18^2/10)^2 / (10 - 1)]`

df ≈ 24.574

To find the critical value, determine the t-value associated with a 90% confidence level and the calculated degrees of freedom.

Using a t-distribution table, the critical value for a 90% confidence level with 24.574 degrees of freedom is approximately 1.711 (rounded to three decimal places).

Therefore, the critical value to be used in constructing the confidence interval is approximately 1.711.