In order to determine the critical value for the hypothesis test, we need to find the upper and lower critical values from the chi-square distribution.
Given:
Sample size (n) = 30
Sample variance (s^2) = 4.41
Population variance (σ^2) = 1.44
To test the hypothesis at the 1% level of significance, we use a chi-square distribution with (n-1) degrees of freedom.
The null hypothesis (H0) is that the population variance is equal to the past temperature variance.
The alternative hypothesis (H1) is that the population variance differs from the past temperature variance.
Since we want to test whether the population variance differs, this is a two-tailed test. Therefore, we need to divide the significance level (α) by 2.
Significance level (α) = 0.01
α/2 = 0.01/2 = 0.005 (for each tail)
The degrees of freedom for the chi-square distribution is (n-1).
df = 30 - 1 = 29
Using a chi-square table or calculator, we can find the upper and lower critical values for the 0.005 significance level with 29 degrees of freedom.
The upper critical value (c1) is the value where the cumulative probability is 0.995.
The lower critical value (c2) is the value where the cumulative probability is 0.005.
For a chi-square distribution with 29 degrees of freedom, the upper critical value (c1) is approximately 46.963.
For the lower critical value (c2), we can use the fact that the chi-square distribution is symmetric. So, c2 = -c1 = -46.963.
Therefore, the critical value for this hypothesis test is c1 = 46.963.
A manufacturer of candy must monitor the temperature at which the sweets are baked. Too much variation will cause inconsistency in the taste of the sweets. Past records shows that the variance of the temperature has been 1.440. A random sample of 30 batches of sweets is selected and the variance of the temperature is 4.41. Testing at the 1% level of significance whether the population variance differs from the past temperature variance showed the critical value is given as:
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