Asked by Shay
A sample of 7 department store sizes (in thousands of square feet) and revenues (in millions of dollars) are found to have a linear coefficient of 0.445. Find the critical values for the linear correlation coefficient, assuming a 0.05 significance level. Is there sufficient evidence to conclude that there is a linear correlation between size and revenue of department stores?
A) Critical values = ±0.875; there is not sufficient evidence to conclude that there is a linear correlation between size and revenue of department stores.
B) Critical values = ±0.754; there is not sufficient evidence to conclude that there is a linear correlation between size and revenue of department stores.
C) Critical values = ±0.754; there is sufficient evidence to conclude that there is a linear correlation between size and revenue of department stores.
D) Critical values = ±0.875; there is sufficient evidence to conclude that there is a linear correlation between size and revenue of department stores.
A) Critical values = ±0.875; there is not sufficient evidence to conclude that there is a linear correlation between size and revenue of department stores.
B) Critical values = ±0.754; there is not sufficient evidence to conclude that there is a linear correlation between size and revenue of department stores.
C) Critical values = ±0.754; there is sufficient evidence to conclude that there is a linear correlation between size and revenue of department stores.
D) Critical values = ±0.875; there is sufficient evidence to conclude that there is a linear correlation between size and revenue of department stores.
Answers
Answered by
GPT-4o mini
To determine the critical values for the linear correlation coefficient at a significance level of 0.05, we first need to find the critical value using the t-distribution, as the correlation coefficient follows a t-distribution with \( n - 2 \) degrees of freedom, where \( n \) is the number of paired observations.
In this case, you have \( n = 7 \), which gives us \( n - 2 = 5 \) degrees of freedom.
1. **Determine the critical t-value** for a one-tailed test at the significance level of 0.05 with 5 degrees of freedom. Using a t-table or calculator, the critical t-value is approximately 2.571.
2. **Calculate the critical correlation coefficient (r)** using the following formula:
\[
r_{critical} = \sqrt{\frac{t^2}{t^2 + n - 2}}
\]
Plugging in our critical t-value:
\[
r_{critical} = \sqrt{\frac{2.571^2}{2.571^2 + 5}} \approx \sqrt{\frac{6.607}{11.607}} \approx \sqrt{0.569} \approx 0.754
\]
Thus, the critical values for the correlation coefficient at the 0.05 significance level are approximately ±0.754.
3. **Evaluate the linear correlation coefficient of 0.445** against the critical values:
Since 0.445 is less than 0.754, we do not reject the null hypothesis and conclude that there is insufficient evidence to support a linear correlation between the size and revenue of department stores.
Therefore, the correct answer is:
**B) Critical values = ±0.754; there is not sufficient evidence to conclude that there is a linear correlation between size and revenue of department stores.**
In this case, you have \( n = 7 \), which gives us \( n - 2 = 5 \) degrees of freedom.
1. **Determine the critical t-value** for a one-tailed test at the significance level of 0.05 with 5 degrees of freedom. Using a t-table or calculator, the critical t-value is approximately 2.571.
2. **Calculate the critical correlation coefficient (r)** using the following formula:
\[
r_{critical} = \sqrt{\frac{t^2}{t^2 + n - 2}}
\]
Plugging in our critical t-value:
\[
r_{critical} = \sqrt{\frac{2.571^2}{2.571^2 + 5}} \approx \sqrt{\frac{6.607}{11.607}} \approx \sqrt{0.569} \approx 0.754
\]
Thus, the critical values for the correlation coefficient at the 0.05 significance level are approximately ±0.754.
3. **Evaluate the linear correlation coefficient of 0.445** against the critical values:
Since 0.445 is less than 0.754, we do not reject the null hypothesis and conclude that there is insufficient evidence to support a linear correlation between the size and revenue of department stores.
Therefore, the correct answer is:
**B) Critical values = ±0.754; there is not sufficient evidence to conclude that there is a linear correlation between size and revenue of department stores.**
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