Consider the following hypothesis test:

[tex]\[
\begin{array}{l}
H_0: \mu_1 - \mu_2 \leq 0 \\
H_a: \mu_1 - \mu_2 \ \textgreater \ 0
\end{array}
\][/tex]

The following results are for two independent samples taken from the two populations:

[tex]\[
\begin{array}{ll}
\text{Sample 1} & \text{Sample 2} \\
n_1 = 40 & n_2 = 45 \\
\bar{x}_1 = 25.1 & \bar{x}_2 = 23.0 \\
\sigma_1 = 4.8 & \sigma_2 = 6.0
\end{array}
\][/tex]

a. What is the value of the test statistic (round to 2 decimals)?
[tex]\[
\square
\][/tex]

b. What is the [tex]$p$[/tex]-value (round to 4 decimals)? Use [tex]$z$[/tex]-value rounded to 2 decimal places.
[tex]\[
\square
\][/tex]

c. With [tex]$\alpha = 0.05$[/tex], what is your hypothesis testing conclusion?
[tex]\[
\text{p-value is } \square \quad \text{Select your answer:} \quad H_0 \text{ or } H_a
\][/tex]



Answer :

Certainly! Let's step through each part of the hypothesis test with the given information:

### Given Data:
- Sample sizes:
[tex]\( n_1 = 40 \)[/tex]
[tex]\( n_2 = 45 \)[/tex]

- Sample means:
[tex]\( \bar{x}_1 = 25.1 \)[/tex]
[tex]\( \bar{x}_2 = 23.0 \)[/tex]

- Population standard deviations:
[tex]\( \sigma_1 = 4.8 \)[/tex]
[tex]\( \sigma_2 = 6.0 \)[/tex]

- Significance level:
[tex]\( \alpha = 0.05 \)[/tex]

### Hypothesis:
[tex]\[ \begin{aligned} H_0: \mu_1 - \mu_2 \leq 0 \\ H_a: \mu_1 - \mu_2 > 0 \end{aligned} \][/tex]

### a. Calculation of the Test Statistic (z-value):
The test statistic (z-value) for two independent samples can be calculated using the formula:
[tex]\[ z = \frac{\bar{x}_1 - \bar{x}_2}{\sqrt{\left(\frac{\sigma_1^2}{n_1}\right) + \left(\frac{\sigma_2^2}{n_2}\right)}} \][/tex]

From the given data:
[tex]\[ \sigma_1 = 4.8, \quad n_1 = 40, \quad \sigma_2 = 6.0, \quad n_2 = 45 \][/tex]

First, calculate the standard error (SE):
[tex]\[ SE = \sqrt{\left(\frac{\sigma_1^2}{n_1}\right) + \left(\frac{\sigma_2^2}{n_2}\right)} = \sqrt{\left(\frac{4.8^2}{40}\right) + \left(\frac{6.0^2}{45}\right)} \][/tex]

Then, calculate:
[tex]\[ z = \frac{25.1 - 23.0}{SE} \][/tex]

Using this formula, we obtain:
[tex]\[ z \approx 1.79 \][/tex]

So, the value of the test statistic is:
[tex]\[ \boxed{1.79} \][/tex]

### b. Calculation of the p-value:
The p-value can be determined by finding the area to the right of the z-value in the standard normal distribution.

For [tex]\( z = 1.79 \)[/tex], the p-value is the area to the right of [tex]\( z = 1.79 \)[/tex].

The p-value comes out to be approximately:
[tex]\[ \boxed{0.0367} \][/tex]

### c. Hypothesis Testing Conclusion:
With [tex]\(\alpha = 0.05\)[/tex]:
- We compare the p-value to the significance level [tex]\(\alpha\)[/tex]:
- If [tex]\( p \text{-value} < \alpha \)[/tex], reject [tex]\( H_0 \)[/tex].
- If [tex]\( p \text{-value} \geq \alpha \)[/tex], do not reject [tex]\( H_0 \)[/tex].

In this case:
[tex]\[ p \text{-value} = 0.0367 < 0.05 \][/tex]

Since the p-value is less than the significance level [tex]\( \alpha \)[/tex], we reject the null hypothesis [tex]\( H_0 \)[/tex].

Thus, our conclusion with [tex]\( \alpha = 0.05 \)[/tex] is:
[tex]\[ \boxed{\text{Reject } H_0} \][/tex]