Question
In a survey, 30 random customers who purchased concessions at a movie theater listed the money they spent. The standard deviation was found to be $1.28. Find the critical value(s) needed to test a claim that the standard deviation of all individual money spent at the concession stand at movie theaters is equal to $1.45. Use a 0.05 significance level.
A) 16.047
B) 17.708, 42.557
C) 17.708
D) 16.047, 45.722
A) 16.047
B) 17.708, 42.557
C) 17.708
D) 16.047, 45.722
Answers
GPT-4o mini
To find the critical values needed for testing the claim about the standard deviation, we can use the chi-square distribution. The test is based on the null hypothesis \( H_0: \sigma = \sigma_0 \) (where \( \sigma_0 \) is the claimed standard deviation, in this case, $1.45) and the alternative hypothesis \( H_a: \sigma \neq \sigma_0 \).
Given:
- Sample standard deviation \( s = 1.28 \)
- Claimed population standard deviation \( \sigma_0 = 1.45 \)
- Sample size \( n = 30 \)
- Significance level \( \alpha = 0.05 \)
To find the critical values, we need the chi-square statistic, which is calculated using the following formula:
\[
\chi^2 = \frac{(n-1)s^2}{\sigma_0^2}
\]
Where:
- \( n-1 = 29 \)
- \( s^2 = (1.28)^2 \)
- \( \sigma_0^2 = (1.45)^2 \)
Calculating \( s^2 \) and \( \sigma_0^2 \):
\[
s^2 = (1.28)^2 = 1.6384
\]
\[
\sigma_0^2 = (1.45)^2 = 2.1025
\]
Now plugging these values into the chi-square formula:
\[
\chi^2 = \frac{(29)(1.6384)}{2.1025} \approx \frac{47.499}{2.1025} \approx 22.612
\]
Next, we will determine the critical values from the chi-square distribution for \( n - 1 = 29 \) degrees of freedom at a significance level of 0.05 for a two-tailed test. We will look up the critical values in the chi-square distribution table or use a calculator:
1. **Lower tail critical value**: \( \chi^2_{\alpha/2, 29} \) (0.025 in the lower tail)
2. **Upper tail critical value**: \( \chi^2_{1-\alpha/2, 29} \) (0.975 in the upper tail)
From the chi-square distribution table:
- The critical value for \( \alpha/2 = 0.025 \) and 29 degrees of freedom is approximately **16.047**.
- The critical value for \( 1-\alpha/2 = 0.975 \) and 29 degrees of freedom is approximately **45.722**.
Hence, the critical values are \( 16.047 \) and \( 45.722 \).
Therefore, the correct answer is:
**D) 16.047, 45.722**
Given:
- Sample standard deviation \( s = 1.28 \)
- Claimed population standard deviation \( \sigma_0 = 1.45 \)
- Sample size \( n = 30 \)
- Significance level \( \alpha = 0.05 \)
To find the critical values, we need the chi-square statistic, which is calculated using the following formula:
\[
\chi^2 = \frac{(n-1)s^2}{\sigma_0^2}
\]
Where:
- \( n-1 = 29 \)
- \( s^2 = (1.28)^2 \)
- \( \sigma_0^2 = (1.45)^2 \)
Calculating \( s^2 \) and \( \sigma_0^2 \):
\[
s^2 = (1.28)^2 = 1.6384
\]
\[
\sigma_0^2 = (1.45)^2 = 2.1025
\]
Now plugging these values into the chi-square formula:
\[
\chi^2 = \frac{(29)(1.6384)}{2.1025} \approx \frac{47.499}{2.1025} \approx 22.612
\]
Next, we will determine the critical values from the chi-square distribution for \( n - 1 = 29 \) degrees of freedom at a significance level of 0.05 for a two-tailed test. We will look up the critical values in the chi-square distribution table or use a calculator:
1. **Lower tail critical value**: \( \chi^2_{\alpha/2, 29} \) (0.025 in the lower tail)
2. **Upper tail critical value**: \( \chi^2_{1-\alpha/2, 29} \) (0.975 in the upper tail)
From the chi-square distribution table:
- The critical value for \( \alpha/2 = 0.025 \) and 29 degrees of freedom is approximately **16.047**.
- The critical value for \( 1-\alpha/2 = 0.975 \) and 29 degrees of freedom is approximately **45.722**.
Hence, the critical values are \( 16.047 \) and \( 45.722 \).
Therefore, the correct answer is:
**D) 16.047, 45.722**