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Question
Math
Posted 6 months ago

Janine is studying the relationship between the size of a diamond (in carats) and its price. She obtains size and price data for a random sample of 64 diamonds. Here is computer output from a least-squares regression analysis on her sample:
\begin{tabular}{lrrrr} 
Predictor & Coef & SE Coef & T & P \\
\hline Constant & 8.00 & 60.76 & 0.13 & 0.90 \\
Carat & 2722.34 & 259.92 & 10.47 & 0.00 \\
S =132.30=132.30 & R-sq =63.9%=63.9 \% & & &
\end{tabular}

Assume that all conditions for inference have been met.
Which of these is a 95%95 \% confidence interval for the slope of the least squares regression line?

Choose 1 answer:
(A) 2722.34±1.67(259.92)2722.34 \pm 1.67(259.92)
(B) 2722.34±1.96(259.92)2722.34 \pm 1.96(259.92)
(c) 2722.34±2.00(259.92)2722.34 \pm 2.00(259.92)
() 2722.34±1.67(259.9264)2722.34 \pm 1.67\left(\frac{259.92}{\sqrt{64}}\right)
() 2722.34±2.00(259.9264)2722.34 \pm 2.00\left(\frac{259.92}{\sqrt{64}}\right)
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Answer from Sia
Posted 6 months ago
Solution by Steps
step 2
Given that the sample size is 64, we have 63 degrees of freedom. For a 95% confidence interval and 63 degrees of freedom, the t t^* value is approximately 1.96
step 3
The standard error of the slope (SE Coef for Carat) is given as 259.92
step 4
The coefficient for Carat is the slope of the regression line, which is 2722.34
step 5
Using the values from steps 2, 3, and 4, we calculate the confidence interval: 2722.34±1.96×259.92 2722.34 \pm 1.96 \times 259.92
step 6
The calculation for the margin of error is: 1.96×259.92=509.4448 1.96 \times 259.92 = 509.4448
step 7
The lower bound of the confidence interval is: 2722.34509.4448=2212.8952 2722.34 - 509.4448 = 2212.8952
step 8
The upper bound of the confidence interval is: 2722.34+509.4448=3231.7848 2722.34 + 509.4448 = 3231.7848
step 9
Therefore, the 95% confidence interval for the slope is approximately (2212.90,3231.78) (2212.90, 3231.78)
[question number] Answer
B
Key Concept
Confidence Interval for Slope
Explanation
The 95% confidence interval for the slope of a regression line is calculated using the formula: slope ± t* × (SE of slope), where t* is the critical value from the t-distribution for the given confidence level and degrees of freedom.

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