To test the claim that the population standard deviation of potassium content in vitamin tablets is equal to 3.2 mg, we can use the chi-square test for variance.
Step 1: Define Hypotheses
- Null hypothesis (\(H_0\)): \(\sigma = 3.2\) mg (the standard deviation of the potassium content is equal to 3.2 mg).
- Alternative hypothesis (\(H_a\)): \(\sigma \neq 3.2\) mg (the standard deviation is not equal to 3.2 mg).
Step 2: Determine the Sample Statistics
Given:
- Sample size (\(n\)) = 25
- Sample standard deviation (\(s\)) = 3.7 mg
- Hypothesized population standard deviation (\(\sigma_0\)) = 3.2 mg
Step 3: Calculate the Test Statistic
The test statistic for the chi-square test is calculated using the formula:
\[ \chi^2 = \frac{(n-1)s^2}{\sigma_0^2} \]
Plugging in the values:
- \(n - 1 = 25 - 1 = 24\)
- \(s^2 = (3.7)^2 = 13.69\)
- \(\sigma_0^2 = (3.2)^2 = 10.24\)
\[ \chi^2 = \frac{(24)(13.69)}{10.24} = \frac{328.56}{10.24} \approx 32.1 \]
Step 4: Degrees of Freedom
The degrees of freedom \(df = n - 1 = 24\).
Step 5: Determine Critical Values
At the 0.05 significance level for a two-tailed test, we look for the critical values in the chi-square distribution with 24 degrees of freedom.
Using a chi-square table or calculator:
- The critical value for \( \chi^2_{0.025, 24} \) (upper tail) is approximately 39.364.
- The critical value for \( \chi^2_{0.975, 24} \) (lower tail) is approximately 12.401.
Conclusion
The critical values are:
- Lower tail: 12.401
- Upper tail: 39.364
Thus, the answer is:
A) 12.401, 39.364