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To determine whether the given statistical series follows a normal distribution using the chi-square (χ²) test at significance levels of 1% and 5%, we can break down the steps as follows:

### Step 1: Define the Hypotheses
- **Null Hypothesis (H₀):** The data follows a normal distribution.
- **Alternative Hypothesis (H₁):** The data does not follow a normal distribution.

### Step 2: Calculate the Expected Frequencies
For a normal distribution, we need to calculate the expected frequency for each class (MPa range) based on the total sample size (49) and the estimated mean (μ) and standard deviation (σ) of the sample.

The observed frequencies are already provided in the table.

### Step 3: Compute the Chi-Square Statistic
The chi-square test statistic is calculated as:

\[
\chi^2 = \sum \frac{(O_i - E_i)^2}{E_i}
\]

Where:
- \(O_i\) = observed frequency in the ith class
- \(E_i\) = expected frequency in the ith class

The sum is taken over all classes.

### Step 4: Compare to Critical Value
To decide whether to reject the null hypothesis, we compare the calculated \(\chi^2\) statistic to the critical value from the chi-square distribution table for the appropriate degrees of freedom (\(df = k - p - 1\), where \(k\) is the number of classes, and \(p\) is the number of parameters estimated, typically 2 for the mean and standard deviation).

Let me know if you'd like me to calculate the expected frequencies or need further clarifications on these steps.

Would you like more details or calculations? Here are 5 related questions you might explore:

1. How do you calculate the mean and standard deviation for this dataset?
2. What are the degrees of freedom in this specific chi-square test?
3. How do you determine the expected frequencies for each class?
4. What is the interpretation if the chi-square statistic exceeds the critical value?
5. How would the decision change between a 1% and 5% significance level?

**Tip:** The chi-square test is particularly sensitive to small expected frequencies; if the expected frequency in any class is too small, it may affect the accuracy of the test.

Does this statistical series follow a normal distribution? Justify your answer using the chi-square test at (i) the 1% significance level and (ii) the 5% significance level.

Learn how to apply the chi-square test to determine if a statistical series follows a normal distribution. This problem includes calculating expected frequencies, setting hypotheses, and using degrees of freedom. Explore how decisions differ at 1% and 5% significance levels.

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