Sure, let's go through this step-by-step.
### Step 1: Set up the hypotheses and state the sample size requirement
- Null Hypothesis [tex]\((H_0)\)[/tex]: The population mean is 24.
[tex]\(H_0: \mu = 24\)[/tex]
- Alternative Hypothesis [tex]\((H_a)\)[/tex]: The population mean is not 24.
[tex]\(H_a: \mu \neq 24\)[/tex]
The sample size requirement is met because the sample size is 35. If the sample size is at least 30, the Central Limit Theorem allows us to use the normal distribution for the test.
### Step 2: Identify the correct distribution to use and the value of [tex]\(\alpha\)[/tex]
Since the population standard deviation is known, we will use a Z-test.
- The value of [tex]\(\alpha\)[/tex] (significance level) is 0.10.
### Step-by-Step Test Procedure
1. Calculate the test statistic (Z):
[tex]\[
Z = \frac{\bar{X} - \mu_0}{\sigma / \sqrt{n}}
\][/tex]
Here, [tex]\(\bar{X}\)[/tex] is the sample mean (23.4), [tex]\(\mu_0\)[/tex] is the population mean under the null hypothesis (24), [tex]\(\sigma\)[/tex] is the population standard deviation (4), and [tex]\(n\)[/tex] is the sample size (35).
Plugging in the values:
[tex]\[
Z = \frac{23.4 - 24}{4 / \sqrt{35}} \approx -0.89
\][/tex]
2. Determine the critical Z values for a two-tailed test at [tex]\(\alpha = 0.10\)[/tex]:
Since [tex]\(\alpha = 0.10\)[/tex], each tail will have [tex]\(\alpha/2 = 0.05\)[/tex].
The critical Z values are:
[tex]\[
Z_{critical, low} = -1.64 \quad \text{and} \quad Z_{critical, high} = 1.64
\][/tex]
3. Calculate the p-value:
The p-value is the probability that the test statistic is as extreme as, or more extreme than, the observed value under the null hypothesis.
[tex]\[
p\text{-value} = 2 \times \text{P}(Z < -0.89) \approx 2 \times 0.18745 = 0.3749
\][/tex]
4. Make a decision:
- Compare the p-value with [tex]\(\alpha\)[/tex]: [tex]\(0.3749 > 0.10\)[/tex]
- Since the p-value is greater than [tex]\(\alpha\)[/tex], we fail to reject the null hypothesis.
Hence, there is not enough evidence to reject the null hypothesis at the 10% significance level. The data do not suggest that the population mean is different from 24.
