To test the claim that $\mu \neq 3.5$, we perform a hypothesis test with the following setup:

1. Null Hypothesis ($H_0$):

$H_0: \mu = 3.5$

2. Alternative Hypothesis ($H_a$):

$H_a: \mu \neq 3.5$

This is a two-tailed test because the alternative hypothesis does not specify a direction (i.e., $\mu$ could be either greater than or less than 3.5).

3. Level of Significance ($\alpha$):

$\alpha = 0.05$

Since this is a two-tailed test, we split the significance level between the two tails of the distribution. Thus, each tail will have $\alpha/2 = 0.025$.

4. Critical Value Approach:

To reject the null hypothesis, the test statistic must fall in the rejection region, which is determined by the critical values from the standard normal distribution (assuming that the sample size is large enough for the Central Limit Theorem to apply).

For a two-tailed test with $\alpha = 0.05$, the critical values from the standard normal distribution ($z$-distribution) are: $z_{\alpha/2} = \pm 1.96$ So, the rejection regions are: $z < -1.96 \quad \text{or} \quad z > 1.96$

5. Decision Rule:

Calculate the test statistic ($z$-score) using the formula: $z = \frac{\bar{x} - \mu_0}{\sigma/\sqrt{n}}$ where:

• $\bar{x}$ is the sample mean,
• $\mu_0 = 3.5$ is the hypothesized population mean,
• $\sigma$ is the population standard deviation (or the sample standard deviation if the population standard deviation is unknown),
• $n = 48$ is the sample size.

If the computed $z$-value is less than $-1.96$ or greater than $1.96$, you reject the null hypothesis.

Conclusion:

You should reject $H_0$ if the test statistic falls in the rejection regions: $z < -1.96$ or $z > 1.96$.

Would you like more details on how to compute the test statistic, or do you have any other questions?

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Here are 5 related questions to extend your understanding:

1. How do you calculate the test statistic if the population standard deviation is unknown?
2. What is the impact of using a one-tailed test instead of a two-tailed test?
3. How does increasing the sample size affect the hypothesis test outcome?
4. How would the rejection region change if the level of significance was $\alpha = 0.01$?
5. What assumptions must hold for the Central Limit Theorem to apply in this test?

Tip: Always check if the population standard deviation is known, as it changes the approach (Z-test vs T-test).