Question

A simple random sample of size 28 has mean x=7.26 and the population standard deviation is o=3.72. The population is not approximately normal. Can you conclude that the population mean differs from 9?

Answer 1

To determine if the population mean differs from 9, we can perform a hypothesis test. Specifically, we will use a one-sample z-test since we know the population standard deviation. Here's a step-by-step breakdown of the process:

### Step 1: Set Up Hypotheses

- Null Hypothesis (H₀): μ = 9 (the population mean is equal to 9)
- Alternative Hypothesis (H₁): μ ≠ 9 (the population mean is not equal to 9)

### Step 2: Calculate the Test Statistic

We'll use the z-test statistic formula:

\[
z = \frac{\bar{x} - \mu_0}{\sigma / \sqrt{n}}
\]

Where:
- \(\bar{x}\) = sample mean = 7.26
- \(\mu_0\) = hypothesized population mean = 9
- \(\sigma\) = population standard deviation = 3.72
- \(n\) = sample size = 28

Plugging in the values:

\[
z = \frac{7.26 - 9}{3.72 / \sqrt{28}}
\]

First, calculate the standard error (SE):

\[
SE = \frac{3.72}{\sqrt{28}} \approx \frac{3.72}{5.2915} \approx 0.702
\]

Now calculate the z-value:

\[
z = \frac{7.26 - 9}{0.702} \approx \frac{-1.74}{0.702} \approx -2.48
\]

### Step 3: Determine the Critical Value and p-value

Given that we are performing a two-tailed test, we need to find the critical z-values for a significance level (α) of 0.05. The critical z-values are approximately ±1.96.

Now let's check the p-value corresponding to our calculated z-value of -2.48. Using z-tables or software, we find:

\[
\text{p-value} \approx 0.013 (two-tailed)
\]

### Step 4: Compare and Conclude

Since the p-value (approximately 0.013) is less than the significance level of α = 0.05, we reject the null hypothesis.

### Conclusion

There is sufficient evidence to conclude that the population mean differs from 9.

Answer 2

The population standard deviation[is or isnt] known.
The sample size n [is or isnt] greater than .
The population [is or isnt] approximately normal.
The correct decision is to [use the z-test, use the t-test, or consult a statistician?].

Answer 3

Let's analyze the points you've mentioned based on the information provided.

1. **The population standard deviation is known**: **is**
- You provided the population standard deviation \(\sigma = 3.72\).

2. **The sample size \(n\) is greater than 30**: **isn't**
- You mentioned that the sample size \(n = 28\), which is less than 30.

3. **The population is approximately normal**: **isn't**
- It was specified that the population is not approximately normal.

4. **The correct decision is to**: **use the t-test**
- Since the population is not approximately normal and our sample size is less than 30, it is more appropriate to use the t-test for hypothesis testing in this case.

In summary:

- The population standard deviation **is** known.
- The sample size \(n\) **isn't** greater than 30.
- The population **isn't** approximately normal.
- The correct decision is to **use the t-test**.