Question

Math

Posted 6 months ago

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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 $=132.30$ & R-sq $=63.9 \%$ & & &
\end{tabular}
Assume that all conditions for inference have been met.
Which of these is a $95 \%$ confidence interval for the slope of the least squares regression line?
Choose 1 answer:
(A) $2722.34 \pm 1.67(259.92)$
(B) $2722.34 \pm 1.96(259.92)$
(c) $2722.34 \pm 2.00(259.92)$
() $2722.34 \pm 1.67\left(\frac{259.92}{\sqrt{64}}\right)$
() $2722.34 \pm 2.00\left(\frac{259.92}{\sqrt{64}}\right)$
```

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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 $t^*$ 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: $2722.34 \pm 1.96 \times 259.92$

step 6

The calculation for the margin of error is: $1.96 \times 259.92 = 509.4448$

step 7

The lower bound of the confidence interval is: $2722.34 - 509.4448 = 2212.8952$

step 8

The upper bound of the confidence interval is: $2722.34 + 509.4448 = 3231.7848$

step 9

Therefore, the 95% confidence interval for the slope is approximately $(2212.90, 3231.78)$

[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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