Let's break this down and solve it step by step. Here’s what we are tasked with:

Hypothesis Testing Problem Details:

1. Null Hypothesis $H_0$: $\mu = 30$
2. Alternative Hypothesis $H_1$: $\mu < 30$ (left-tailed test)
3. Sample size $n = 45$
4. Sample mean $\bar{x} = 26$
5. Population standard deviation $\sigma = 9$
6. Test significance levels:
   • $\alpha = 0.05$
   • $\alpha = 0.01$

(a) Compute the test statistic $z$:

The formula for $z$ in a hypothesis test is:

$z = \frac{\bar{x} - \mu}{\frac{\sigma}{\sqrt{n}}}$

Substitute the values:

• $\bar{x} = 26$, $\mu = 30$, $\sigma = 9$, $n = 45$

$z = \frac{26 - 30}{\frac{9}{\sqrt{45}}}$

1. Compute the denominator: $\frac{\sigma}{\sqrt{n}} = \frac{9}{\sqrt{45}} = \frac{9}{6.708} \approx 1.342$

2. Compute $z$: $z = \frac{-4}{1.342} \approx -2.98$

(b) Is $H_0$ rejected at $\alpha = 0.05$?

For a left-tailed test at $\alpha = 0.05$:

• The critical value is $z_\text{critical} = -1.645$.

Since $z = -2.98$ is less than $z_\text{critical}$, we reject $H_0$ at $\alpha = 0.05$.

(c) Is $H_0$ rejected at $\alpha = 0.01$?

For a left-tailed test at $\alpha = 0.01$:

• The critical value is $z_\text{critical} = -2.33$.

Since $z = -2.98$ is less than $z_\text{critical}$, we reject $H_0$ at $\alpha = 0.01$.

Summary of Results:

1. Test statistic $z$: $-2.98$
2. $H_0$ is rejected at both $\alpha = 0.05$ and $\alpha = 0.01$.

Would you like further details or explanations on hypothesis testing?

Related Questions:

1. How do you calculate the test statistic for different distributions (e.g., t-distribution)?
2. What happens if $\sigma$ is unknown? When do we use a t-test instead of a z-test?
3. How do you interpret p-values in hypothesis testing?
4. What is the difference between one-tailed and two-tailed tests?
5. Can you explain the assumptions behind using a z-test?

Tip: Always check the sample size and population standard deviation when deciding between a z-test and a t-test.