Introduction To Statistics And Data Analysis
Introduction To Statistics And Data Analysis
6th Edition
ISBN: 9781337793612
Author: PECK, Roxy.
Publisher: Cengage Learning,
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Chapter 13.2, Problem 19E

An experiment to study the relationship between x = Time spent exercising (minutes) and y = Amount of oxygen consumed during the exercise period resulted in the following summary statistics.

n = 20 x = 50 y = 16 , 705 x 2 = 150 y 2 = 14 , 194 , 231 x y = 44 , 194

  1. a. Estimate the slope and y intercept of the population regression line.
  2. b. One sample observation on oxygen usage was 757 for a 2-minute exercise period. What amount of oxygen consumption is predicted for this exercise period, and what is the corresponding residual?
  3. c. Calculate a 99% confidence interval for the average change in oxygen consumption associated with a 1-minute increase in exercise time.

a.

Expert Solution
Check Mark
To determine

Estimate the slope.

Find the y intercept of the population regression equation.

Answer to Problem 19E

The slope is 97.26.

The y intercept of the population regression equation is 592.1.

Explanation of Solution

Calculation:

The given information is that, the variable x denotes the time spent exercising (minutes) and y denotes amount of oxygen consumed during the exercise period. The summary statistics are,

n=20,x=50,y=16,705,x2=150,y2=14,194,231,xy=44,194

Formula for Slope:

b=SxySxx

In the formula, Sxy=xy(x)(y)n and Sxx=x2(x)2n.

Formula for intercept:

a=y¯bx¯

In the formula y¯=yn,x¯=xn

Sxy:

Substitute xy=44,194,x=50,y=16,705,n=20

Sxy=44,194(50)(16,705)20=44,194835,25020=44,19441,762.5=2,431.5

Sxx:

Substitute n=20,x=50,x2=150

Sxx=x2(x)2n=150(50)220=1502,50020=150125=25

Slope:

Substitute, Sxy=2,431.5,Sxx=25

b=2,431.525=97.26

Hence, the slope is 97.26.

y¯:

Substitute n=20,y=16,705 in the formula.

y¯=16,70520=835.25

x¯:

Substitute n=20,x=50 in the formula.

x¯=5020=2.5

Intercept:

Substitute y¯=835.25,b=97.26,x¯=2.5 in the formula

a=835.25(97.26)2.5=835.25243.15=592.1

Hence, the y intercept of the population regression equation is 592.1.

b.

Expert Solution
Check Mark
To determine

Find the amount of oxygen consumption that is predicted for 2-minute exercise period.

Find the corresponding residual.

Answer to Problem 19E

The amount of oxygen consumption that is predicted for 2-minute exercise period is 786.62.

The corresponding residual is –29.62.

Explanation of Solution

Calculation:

The given information is that for a 2 minute period the oxygen intake was 757. From part (a), the value of slope is b=97.26 and intercept is a=592.1.

The estimated regression line is given by, y=a+bx

Substitute, b=97.26 and α=592.1 in the regression equation

y=592.1+97.26x

Predicted value:

Substitute x=2 in the estimated regression equation.

y=592.1+97.26(2)=592.1+194.52=786.62

Hence, the amount of oxygen consumption that is predicted for 2-minute exercise period is 786.62.

Residual:

The formula for residual is,

Residual=yy^

Substitute, y=757,y^=786.62 in the formula.

y=757786.62=29.62

Hence, the corresponding residual is –29.62.

c.

Expert Solution
Check Mark
To determine

Find 99% confidence interval for the average change in oxygen consumption associated with a 1-minute increase in exercise time.

Answer to Problem 19E

The 99% confidence interval for the average change in oxygen consumption associated with a 1-minute increase in exercise time is (87.7641,106.7559).

Explanation of Solution

Calculation:

The given information is that, the variable x denotes the time spent exercising (minutes) and y denotes amount of oxygen consumed during the exercise period. The level of significance is 0.01. The summary statistics are,

n=20,x=50,y=16,705,x2=150,y2=14,194,231,xy=44,194

Formula for Degrees of freedom:

The formula for degrees of freedom is,

df=n2

In the formula, n is the total number of observations.

Formula for estimated standard deviation of statistic b:

sb=seSxx

In the formula, se=SSResidn2, SSResid=y2aybxy, n denotes the total number of observations and Sxx=x2(x)2n

Formula for Confidence interval for β:

Confidence interval=b±(tcriticalvalue)sb

In the formula b denotes the estimated slope and sb denotes the estimated standard deviation of statistic b.

Degrees of freedom:

Substitute n=20 in the degrees of freedom formula.

df=202=18

From the Appendix: Table 3 the t Critical Values:

  • Locate the value 18 in the degrees of freedom (df) column.
  • Locate the 0.99 in the row of central area captured.
  • The intersecting value that corresponds to the 18 with confidence level 0.99 is ±2.88

Thus, the critical value for df=18 with two-tailed test is ±2.88.

SSResid:

Substitute, y2=14,194,231,y=16,705,a=592.1,b=97.26,xy=44,194 in the formula.

SSResid=14,194,231592.1(16,705)97.26(44,194)=14,194,2319,891,030.54,298,308.44=4,892.06

se:

Substitute, SSResid=4,892.06,n=20 in the formula.

se=4,892.06202=4,892.0618=271.7811=16.4858

sb:

Substitute, se=16.4858,Sxx=25 in the formula.

sb=16.485825=16.48585=3.2972

Confidence interval:

Substitute, b=97.26,sb=3.2972,tcritical value=2.88 in the confidence interval formula.

Confidence interval=97.26±(2.88)3.2972=97.26±9.4959=(87.7641,106.7559)

Hence, the 99% confidence interval for β is (87.7641,106.7559).

Interpretation:

The 99% confidence interval is interpreted as, there is 99% confidence that the slope for the population regression line or the average change in oxygen consumption associated with a 1-minute increase in exercise time lies between 87.7641 and 106.7559.

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Chapter 13 Solutions

Introduction To Statistics And Data Analysis

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