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Probability and Statistics for Eng...

9th Edition
Jay L. Devore
Publisher: Cengage Learning
ISBN: 9781305251809

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BuyFindarrow_forward

Probability and Statistics for Eng...

9th Edition
Jay L. Devore
Publisher: Cengage Learning
ISBN: 9781305251809
Chapter 12.4, Problem 49E
Textbook Problem
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You are told that a 95% CI for expected lead content when traffic flow is 15, based on a sample of n = 10 observations, is (462.1, 597.7). Calculate a CI with confidence level 99% for expected lead content when traffic flow is 15.

To determine

Find 99% specified confidence interval for the true mean lead content for 15 traffic flow.

Explanation of Solution

Given info:

The 95% confidence interval for the true mean lead content when the traffic flow is 15 for a sample of 10 observations is (462.1,597.7).

Calculation:

Linear regression model:

A linear regression model is given as y^=b0+b1x where y^ be the predicted values of response variable and x be the predictor variable. b1 be the slope and b0 be the intercept of the line.

A linear regression model is given as y^=β^0+β^1x where y^ be the predicted values of response variable and x be the predictor variable. The β^1 be the estimate of slope and β^0 be the estimate of intercept of the line.

Confidence interval:

The general formula for the (1α)% confidence interval for the conditional mean at x=xp is,

CI=y^p±t(α2,n2)s1n+(xpixin)2Sxx=y^p±t(α2,n2)sY^

Where, y^p be the point estimate for the conditional mean of the response variable at x=xp, and sY^=s×1n+(x*x¯)2Sxx.

Here, the 95% confidence interval for the true mean lead content for 15 traffic flow is (462.1,597.7).

Here, the lower bound is 462.1 and the upper bound is 597.7.

Point estimate of lead content for 15 traffic flow:

Point estimate of lead content for 15 traffic flow is obtained as follows:

p^=Lower bound+Upper bound2p^=462.1+597.72=529.9

Thus, the point estimate of lead content for 15 traffic flow (y^p) is 529.9.

Critical value for 95% confidence level:

For 95% confidence level,

1α=10.95α=0.05α2=0.052=0

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Chapter 12 Solutions

Probability and Statistics for Engineering and the Sciences
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