Stoaches are fictional creatures that feed on funistrada.

According to Wikipedia, adult stoach weights are normally distributed, with a mean of 452 g and a standard deviation of 61 g.

You weigh a random sample of 48 stoaches, and find a sample mean weight of 462.8 g.

Assume Wikipedia is correct about the standard deviation. What is the 𝑝
-value for a two-sided hypothesis test that Wikipedia is also right about the mean weight? (Give your answer to 4 decimal places.)

To find the \( p \)-value for the two-sided hypothesis test, we'll follow these steps:

### Step 1: State the hypotheses
- **Null hypothesis (\( H_0 \))**: The population mean weight of stoaches is 452 g.
- **Alternative hypothesis (\( H_1 \))**: The population mean weight of stoaches is not 452 g.

### Step 2: Calculate the test statistic
The test statistic for the sample mean can be calculated using the following formula for the \( z \)-score:

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

where:
- \( \bar{x} \) is the sample mean (462.8 g),
- \( \mu_0 \) is the hypothesized population mean (452 g),
- \( \sigma \) is the population standard deviation (61 g),
- \( n \) is the sample size (48).

Plugging in the values:

\[
z = \frac{462.8 - 452}{\frac{61}{\sqrt{48}}}
\]

### Step 3: Calculate the z-value
We can now calculate this z-value.

\[
z = \frac{10.8}{\frac{61}{\sqrt{48}}}
\]

Let's calculate it.The calculated \( z \)-value is approximately \( 1.2266 \).

### Step 4: Calculate the \( p \)-value
For a two-sided test, the \( p \)-value is calculated as:

\[
p = 2 \times P(Z > |z|)
\]

We'll now find the \( p \)-value using the standard normal distribution table or by calculating it directly.The \( p \)-value for the two-sided hypothesis test is approximately \( 0.2200 \).

### Summary
Given a \( p \)-value of 0.2200, there is not enough evidence to reject the null hypothesis at common significance levels (e.g., 0.05). This suggests that the sample mean does not provide strong evidence against the hypothesis that the population mean is 452 g.

Do you want further details or have any questions?

Here are 5 related questions:
1. How would the \( p \)-value change if the sample size increased?
2. What is the relationship between the \( p \)-value and the significance level in hypothesis testing?
3. How do we interpret a \( p \)-value in the context of hypothesis testing?
4. What would the \( p \)-value be if the sample mean were 455 g instead?
5. How does the standard deviation affect the \( z \)-score in this test?

**Tip:** When interpreting \( p \)-values, remember that a low \( p \)-value indicates strong evidence against the null hypothesis, while a high \( p \)-value suggests weak evidence.

Learn how to calculate the p-value for a two-sided hypothesis test using normal distribution with a given sample mean and standard deviation. Understand the concepts of hypothesis testing, z-score, and normal distribution. Suitable for advanced students and professionals in statistics and data analysis.

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