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The text in the image is a Japanese statistics problem related to hypothesis testing. Here is the translated text and the steps to solve it:

### Translation:
A cola manufacturer bases the content of each cola bottle on a standard of 500 ml. It is known that the standard deviation of the content is 7 ml. When 100 bottles of cola from this manufacturer were randomly sampled and measured, the average content was found to be 498.5 ml. At a 5% significance level, determine if it can be concluded that the average content per bottle is not 500 ml.

### Solution:

We will perform a hypothesis test using the following steps:

1. **State the Hypotheses:**
   - Null Hypothesis (\(H_0\)): The average content per bottle is 500 ml. \( \mu = 500 \)
   - Alternative Hypothesis (\(H_1\)): The average content per bottle is not 500 ml. \( \mu \neq 500 \)

2. **Determine the Significance Level:**
   - Significance level (\(\alpha\)) = 0.05

3. **Test Statistic:**
   - Since the population standard deviation is known and the sample size is large (\(n = 100\)), we use the Z-test.
   - The test statistic is calculated as:
     \[
     Z = \frac{\bar{X} - \mu_0}{\frac{\sigma}{\sqrt{n}}}
     \]
     where:
     - \(\bar{X} = 498.5\) ml (sample mean)
     - \(\mu_0 = 500\) ml (hypothesized mean)
     - \(\sigma = 7\) ml (population standard deviation)
     - \(n = 100\) (sample size)

     Substituting the values:
     \[
     Z = \frac{498.5 - 500}{\frac{7}{\sqrt{100}}} = \frac{-1.5}{0.7} \approx -2.14
     \]

4. **Determine the Critical Value:**
   - For a two-tailed test at the 5% significance level, the critical Z-values are \(\pm 1.96\).

5. **Decision:**
   - Compare the calculated Z-value to the critical values.
   - Since \(Z \approx -2.14\) falls outside the range of \(-1.96\) to \(+1.96\), we reject the null hypothesis.

### Conclusion:
At the 5% significance level, there is sufficient evidence to conclude that the average content per bottle is not 500 ml.

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Would you like a detailed explanation of any specific part of the solution?

Here are some related questions you might find interesting:

1. How would the conclusion change if the sample mean was 499 ml instead of 498.5 ml?
2. What is the probability of committing a Type I error in this hypothesis test?
3. How would the result be affected if the sample size was 50 instead of 100?
4. Can you explain what a two-tailed test means in the context of this problem?
5. What assumptions must be satisfied for the Z-test to be valid?
6. How would you calculate the p-value for this test?
7. If the significance level was 1%, would we still reject the null hypothesis?
8. How would the test change if the population standard deviation were unknown?

**Tip:** When conducting hypothesis tests, always check if the sample size is large enough to justify using the Z-test, especially if the population distribution is not normal.

Explore a hypothesis testing problem in statistics: Determine if the average content per cola bottle differs from the standard of 500 ml using a Z-test. Includes step-by-step solution with calculations and interpretation. Suitable for advanced high school students and beyond.

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