To find the critical values for the chi-square distribution based on the given information, we follow these steps:
1. Identify the hypothesis:
- We have an alternative hypothesis [tex]\( H_1 \)[/tex] that states [tex]\( \sigma \neq 9.3 \)[/tex], which indicates that we are dealing with a two-tailed test.
2. Determine the sample size ([tex]\(n\)[/tex]):
- Given [tex]\( n = 28 \)[/tex].
3. Calculate the degrees of freedom (df):
- For a chi-square test, the degrees of freedom ([tex]\(df\)[/tex]) is calculated as [tex]\( n - 1 \)[/tex].
- So, [tex]\( df = 28 - 1 = 27 \)[/tex].
4. Specify the significance level ([tex]\(\alpha\)[/tex]):
- Given [tex]\(\alpha = 0.05\)[/tex]. Since it is a two-tailed test, we will split the significance level into two equal parts: [tex]\(\alpha / 2 = 0.025\)[/tex] for each tail.
5. Find the critical values of the chi-square distribution:
- The lower critical value is found by looking up the chi-square value that corresponds to a cumulative probability of [tex]\(\alpha / 2\)[/tex], which is [tex]\( 0.025 \)[/tex].
- The upper critical value is found by looking up the chi-square value that corresponds to a cumulative probability of [tex]\( 1 - \alpha / 2 \)[/tex], which is [tex]\( 0.975 \)[/tex].
After finding the critical values from a chi-square distribution table or using statistical software:
- The lower critical value is approximately [tex]\( 14.573 \)[/tex].
- The upper critical value is approximately [tex]\( 43.194 \)[/tex].
Therefore, the correct pair of critical values for the chi-square test given the data is [tex]\( (14.573, 43.194) \)[/tex].
Hence, the correct answer is:
[tex]\[ 14.573, 43.194 \][/tex]
