Question

Provided below are summary statistics for independent simple random samples from two populations. Use the pooled​ t-test and the pooled​ t-interval procedure to conduct the required hypothesis test and obtain the specified confidence interval.
x1=21, s1=4, n1=12, x2=20, s2=3, n2=15
A. What are the correct hypotheses for a​ right-tailed test?
b. Compute the test statistic.
c. Determine the​ P-value.
B. The 90​% confidence interval is from ____to ____.

Answer 1

Answer:

(a) $H_o:\mu_1 = \mu_2$     $H_a:\mu_1 > \mu_2$

(b) $t = 0.74$

(c) $p =0.2331$

(d) $CI = (-2.095,4.095)$

Step-by-step explanation:

Given

$\bar x_1=21,\ s_1=4,\ n_1=12,\\ \bar x_2=20,\ s_2=3,\ n_2=15$

Solving (a): The hypotheses

The test is right-tailed, means that the alternate hypothesis will contain greater than sign.

So, we have:

$H_o:\mu_1 = \mu_2$

$H_a:\mu_1 > \mu_2$

Solving (b); The test statistic (t)

This is calculated as:

$t = \frac{\bar x_1 - \bar x_2}{\sqrt{\frac{s_1^2(n_1 - 1) + s_2^2(n_2 - 1)}{n_1 + n_2 - 2} * (\frac{1}{n_1} + \frac{1}{n_2})}}$

So, we have:

$t = \frac{21 - 20}{\sqrt{\frac{4^2(12 - 1) + 3^2(15 - 1)}{12 + 15 - 2} * (\frac{1}{12} + \frac{1}{15})}}$

$t = \frac{1}{\sqrt{\frac{302}{25} * (0.15)}}$

$t = \frac{1}{\sqrt{12.08 * 0.15}}$

$t = \frac{1}{\sqrt{1.812}}$

$t = \frac{1}{1.346}$

$t = 0.74$

Solving (c): The P-value

First, we calculate the degrees of freedom

$df = n_1 + n_2 -2$

$df = 12+15 -2$

$df = 25$

Using the t distribution, the p-value is:

$p =TDIST(0.74,25)$

$p =0.2331$

Solving (d): The 90% confidence interval

Calculate significance level

$\alpha = 1 - CI$

$\alpha = 1 - 90\%$

$\alpha = 0.10$

Calculate the t value (t*)

$t^* = (\alpha/2,df)$

$t^* = (0.10/2,25)$

$t^* = (0.05,25)$

$t^* = 1.708$

The confidence interval is calculated using:

$CI = (\bar x - \bar x_2) \± t^* *\sqrt{\frac{s_1^2(n_1 - 1) + s_2^2(n_2 - 1)}{n_1 + n_2 - 2} * (\frac{1}{n_1} + \frac{1}{n_2})}$

$CI = (21 - 20) \± 1.708 *\sqrt{\frac{4^2(12 - 1) + 3^2(15 - 1)}{12 + 15 - 2} * (\frac{1}{12} + \frac{1}{15})}$

$CI = 1 \± 1.708 *1.812$

$CI = 1 \± 3.095$

Split

$CI = 1 - 3.095 \ or\ 1 + 3.095$

$CI = -2.095 \ or\ 4.095$

$CI = (-2.095,4.095)$