Answer :
a)Based on the given sample, there is sufficient reason to conclude that the population standard deviation is not equal to 8 at the 0.05 level of significance.
b)Based on the classical approach, we fail to find sufficient reason to conclude that the population standard deviation is not equal to 8 at the 0.05 level of significance.
a. Solve using the p-value approach:
To determine if the population standard deviation is not equal to 8, we can perform a hypothesis test using the p-value approach.
Step 1: State the null and alternative hypotheses:
Null hypothesis (H0): The population standard deviation (σ) is equal to 8.
Alternative hypothesis (H1): The population standard deviation (σ) is not equal to 8.
Step 2: Determine the test statistic:
In this case, we use the chi-square distribution to test the hypothesis about the population standard deviation. The test statistic is calculated as:
χ² = (n - 1) * s² / σ²
where n is the sample size, s² is the sample variance, and σ² is the hypothesized population variance under the null hypothesis.
In this scenario, n = 51, s² = 37.5, and the hypothesized population variance under the null hypothesis is σ² = 8² = 64.
Calculating the test statistic:
χ² = (51 - 1) * 37.5 / 64 ≈ 29.34
Step 3: Determine the p-value:
The p-value is the probability of obtaining a test statistic as extreme as the observed value, assuming the null hypothesis is true. Since the alternative hypothesis is two-tailed (not equal to), we need to find the area in both tails of the chi-square distribution.
Using a chi-square distribution table or software, we find that the p-value associated with a test statistic of 29.34 with (n - 1) degrees of freedom (50 degrees of freedom in this case) is very close to zero (p-value ≈ 0).
Step 4: Make a decision:
If the p-value is less than the significance level (α = 0.05 in this case), we reject the null hypothesis. Since the p-value is very close to zero (p-value ≈ 0), it is less than 0.05. Therefore, we reject the null hypothesis.
Conclusion:
Based on the given sample, there is sufficient reason to conclude that the population standard deviation is not equal to 8 at the 0.05 level of significance.
b. Solve using the classical approach:
In the classical approach, we compare the test statistic (in this case, the chi-square statistic) to the critical value from the chi-square distribution at the desired level of significance (α = 0.05).
Step 1: State the null and alternative hypotheses:
Null hypothesis (H0): The population standard deviation (σ) is equal to 8.
Alternative hypothesis (H1): The population standard deviation (σ) is not equal to 8.
Step 2: Determine the critical value:
We need to find the critical value from the chi-square distribution table at a significance level of α = 0.05 with (n - 1) degrees of freedom (50 degrees of freedom in this case).
Looking up the critical value in the chi-square distribution table, we find it to be approximately 69.23.
Step 3: Determine the test statistic:
The test statistic was calculated in the p-value approach as χ² ≈ 29.34.
Step 4: Make a decision:
If the test statistic is greater than the critical value, we reject the null hypothesis. Otherwise, we fail to reject the null hypothesis.
Since the test statistic (29.34) is less than the critical value (69.23), we fail to reject the null hypothesis.
Conclusion:
Based on the classical approach, we fail to find sufficient reason to conclude that the population standard deviation is not equal to 8 at the 0.05 level of significance.
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