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Chapter 14: Q. 14.56 (page 570)

Short Answer

Expert verified

The slope of the population regression line is not 0 and hence the variable, xis not useful for predicting the variable, y.

Interpretation: We can be 90%confident that the slope of the population regression line is somewhere between -0.782and 1.282.

Step by step solution

01

Given information

The equation is y^=1.75+0.25x

We have to find that the predicting variable and the confidence level.

02

Determine the hypothesis 

STEP 1: The null and alternative hypotheses are

H0:β1=0( xis not useful for predicting y)

Ha:β1≠0( xis useful for predictingy)

STEP 2: Decide on the significance level, α.

We are to perform the hypothesis test at the 10%significance level, or α=0.10.

Computation table

03

Determine the equation

Sxy=∑xiyi-∑xi∑yi/n=34-(12)(10)/4=34-120/4=34-30=4Sxx=∑xi2-∑xi2/n=52-(12)2/4=52-144/4=52-36=16
04

The total sum of square SST

Syy=∑yi2-∑yi2/n=30-(10)2/4=30-100/4=30-25=5

Since the P-value =0.5527>α=0.10. We do not reject our null hypothesis H0.

At the 10%significance level, the data do not provide sufficient evidence to conclude that the slope of the population regression line is not 0 and hence the variable, xis not useful for predicting the variable, y.

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Most popular questions from this chapter

In this section, we used the statistic b1as a basis for conducting a hypothesis test to decide whether a regression equation is useful for prediction. Identify two other statistics that can be used as a basis for such a test.

In Exercises 14.48-14.57, we repeat the information from Exercises 14.12-14.21.

a. Decide, at the lore significance level, whether the data provide sufficient evidence to conclude that \(x\) is useful for predicting y.

b. Find a 90rk confidence interval for the slope of the population regression line.

y=2.875-0.625x

In Exercises 14.98-14.108, use the technology of your choice to do the following tasks.
a. Decide whether you can reasonably apply the conditional mean and predicted value t-interval procedures to the data. If so, then also do parts (b)-(f)
b. Determine and interpret a point estimate for the conditional mean of the response variable corresponding to the specified value of the predictor variable.
c. Find and interpret a 95% confidence interval for the conditional mean of the response variable corresponding to the specified value of the predictor variable.
d. Determine and interpret the predicted value of the response variable corresponding to the specified value of the predictor variable.
e. Find and interpret a 95% prediction interval for the value of the response variable corresponding to the specified value of the predictor variable.
f. Compare and discuss the differences between the confidence interval that you obtained in part (c) and the prediction interval that you obtained in part (e).

14.98 Birdies and Score. The data from Exercise 14.34 for number of birdies during a tournament and final score of 63 women golfers are on the WeissStats site. Specified value of the predictor variable:12birdies.

a. Obtain a point estimate for the mean tax efficiency of all mutual fund portfolios with6% of their investments in energy securities,

b. Determine a 95%confidence interval for the mean tax efficiency of all mutual fund portfolios with6% of their investments in energy securities.

c. Find the predicted tax efficiency of a mutual fund portfolio with6% of its investments in energy securities.

d. Determine a95%prediction interval for the tax efficiency of a mutual fund portfolio with 6%of its investments in energy securities.

In Exercises 14.12-14.21, we repeat the data and provide the sample regression equations for Exercises 4.48 -4.57.

a. Determine the standard error of the estimate.

b. Construct a residual plot.

c. Construct a normal probability plot of the residuals.

y=1.75+0.25x
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