To conduct a goodness-of-fit test to determine whether a sample appears to have been drawn from a normal distribution, you would typically use the Chi-Square goodness-of-fit test.
Here are the steps you would follow to perform such a test, including how you can assess each interval's probabilities. However, more specifics regarding the sample data and intervals would be needed for precise calculations.
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Define the Null and Alternative Hypotheses:
- Null Hypothesis (H0): The sample is drawn from a normal distribution.
- Alternative Hypothesis (H1): The sample is not drawn from a normal distribution.
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Determine the intervals and calculate expected frequencies: You need to establish bins or intervals. For a normal distribution, you might choose intervals that reflect the mean and standard deviation of the sample.
To find the expected frequency for each interval, calculate the normal probabilities for each of the intervals using the normal distribution's cumulative distribution function (CDF).
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Calculate the observed frequencies: Count how many data points from your sample fall within each interval to obtain observed frequencies.
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Calculate the expected frequencies: Multiply the total sample size (n = 40) by the probability of each interval based on the normal distribution to find expected frequencies.
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Perform the Chi-Square Test: The test statistic is calculated as: \[ \chi^2 = \sum \frac{(O_i - E_i)^2}{E_i} \] where \(O_i\) is the observed frequency and \(E_i\) is the expected frequency for each interval.
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Determine the degrees of freedom: The degrees of freedom for this test are calculated as: \[ df = k - 1 - p \] where \(k\) is the number of bins, and \(p\) is the number of parameters estimated (in the case of a normal distribution, mean and standard deviation). Usually, \(p = 2\).
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Find the critical value and make a decision: Compare the calculated Chi-square statistic to the critical value from the Chi-square distribution with the appropriate degrees of freedom at your desired level of significance (often 0.05).
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Conclusion: If the Chi-square statistic exceeds the critical value, reject the null hypothesis in favor of the alternative.
For the specific probability for each interval, you would need to specify the intervals as well as the parameters of the normal distribution (mean and standard deviation) derived from your sample. Then, applying the normal distribution CDF or using statistical software, you can calculate those probabilities. If you provide the data or additional context, I can assist with specific calculations.