Understandable Statistics: Concepts and Methods
Understandable Statistics: Concepts and Methods
12th Edition
ISBN: 9781337119917
Author: Charles Henry Brase, Corrinne Pellillo Brase
Publisher: Cengage Learning
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Chapter 9.2, Problem 12P

(a)

To determine

Construct a scatter diagram for the data.

(a)

Expert Solution
Check Mark

Answer to Problem 12P

The scatter diagram for data is,

Understandable Statistics: Concepts and Methods, Chapter 9.2, Problem 12P , additional homework tip  1

Explanation of Solution

Calculation:

The variable x denotes the age of a licensed driver in years and y denotes the percentage of all fatal accidents due to failure to yield the right-of-way.

Step by step procedure to obtain scatter plot using MINITAB software is given below:

  • Choose Graph > Scatterplot.
  • Choose Simple. Click OK.
  • In Y variables, enter the column of x.
  • In X variables, enter the column of y.
  • Click OK.

(b)

To determine

Verify the values of x, x2, y, y2, xy and r.

(b)

Expert Solution
Check Mark

Explanation of Solution

Calculation:

The formula for correlation coefficient is,

r=nxy(x)(y)nx2(x)2ny2(y)2

In the formula, n is the sample size.

The values are verified in the table below,

xyx2y2xy
375136925185
478220964376
57103249100570
671644892561072
773059299002310
8743756918493741
x=372y=112x2=24,814y2=3,194xy=8,254

Hence, the values are verified.

The number of data pairs are n=6. The value of correlation r is,

r=nxy(x)(y)nx2(x)2ny2(y)2=6(8,254)(372×112)6(24,814)(372)26(3,194)(112)2=49,52441,66410,5006,620=7,860102.4695×81.3634=7,8608,337.2669=0.943

The value of r is 0.943 this shows that r is not –0.943.

Hence, the value of r is verified as approximately 0.943.

(c)

To determine

Find the value of x¯.

Find the value of y¯.

Find the value of a.

Find the value of b.

Find the equation of the least-squares line.

(c)

Expert Solution
Check Mark

Answer to Problem 12P

The value of x¯ is 62 years.

The value of y¯ is 18.67%.

The value of a is –27.768.

The value of b is 0.749.

The equation of the least-squares line is y^=27.768+0.749x.

Explanation of Solution

Calculation:

From part (b), the values are x=372, y=112, x2=24,814, y2=3,194, xy=8,254 and n=6

The value of x¯ is,

x¯=xn=3726=62

Hence, the value of x¯ is 62 years.

The value of y¯ is,

y¯=yn=1126=18.67

Hence, the value of y¯ is 18.67%.

The value of b is,

b=nxy(x)(y)nx2(x)2=6(8,254)(372×112)6(24,814)(372)2=49,52441,664148,884138,384=7,86010,500=0.749

Hence, the value of b is 0.749.

The value of a is,

a=y¯bx¯=18.67(0.749)62=18.6746.438=27.768

Hence, the value of a is –27.768.

The equation of the least-squares line is,

y^=a+bx=27.768+0.749x

Hence, the equation of the least-squares line is y^=27.768+0.749x.

(d)

To determine

Construct a scatter diagram with least squares line.

Locate the point (x¯,y¯) on the line.

(d)

Expert Solution
Check Mark

Answer to Problem 12P

The scatter diagram with least squares line with point (x¯,y¯) is,

Understandable Statistics: Concepts and Methods, Chapter 9.2, Problem 12P , additional homework tip  2

Explanation of Solution

Calculation:

In the dataset of failure of yield, also include the point (x¯,y¯). That is, (62,18.67).

Step by step procedure to obtain scatter plot using MINITAB software is given below:

  • Choose Graph > Scatterplot.
  • Choose With regression. Click OK.
  • In Y variables, enter the column of x.
  • In X variables, enter the column of y.
  • Click OK.

(e)

To determine

Calculate the value of the coefficient of determination r2.

Mention percentage of the variation in y that can be explained by variation in x.

Mention percentage of the variation in y that cannot be explained by variation in x.

(e)

Expert Solution
Check Mark

Answer to Problem 12P

The value of the coefficient of determination r2 is 0.889.

The percentage of the variation in y that can be explained by variation in x is 88.9%.

The percentage of the variation in y that cannot be explained by variation in x is 11.1%.

Explanation of Solution

Calculation:

Coefficient of determination(r2):

The coefficient of determination (r2) determines the percent of the variation in response variable that is explained by the predictor variables. A larger value of r2 indicates that the model is a good fit.

From part (b), the value of correlation coefficient r is 0.943. The value of r2 is,

r=0.943r2=(0.943)2=0.889

Hence, the value of the coefficient of determination r2 is 0.889.

About 88.9% of the variation in y (percentage of all fatal accidents due to failure to yield the right-of-way) is explained by x (age of a licensed driver in years). Since the value of r2 is large the model is a good fit for the data.

Hence, the percentage of the variation in y that can be explained by variation in x is 92%.

About 11.1% (=100%88.9%) of the variation y (percentage of all fatal accidents due to failure to yield the right-of-way) cannot be explained by x (age of a licensed driver in years).

Hence, the percentage of the variation in y that cannot be explained by variation in x is 11.1%.

(f)

To determine

Find the percentage of all fatal accidents due to failing to yield the right-of-way for 70-year-olds.

(f)

Expert Solution
Check Mark

Answer to Problem 12P

The percentage of all fatal accidents due to failing to yield the right-of-way for 70-year-olds is 24.662%.

Explanation of Solution

Calculation:

From part (c), the equation of the least-squares line is y^=27.768+0.749x.

Substitute x=70 in equation of the least-squares line.

y^=27.768+0.749(70)=27.768+52.43=24.662

Hence, the percentage of all fatal accidents due to failing to yield the right-of-way for 70-year-olds is 24.662%.

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Chapter 9 Solutions

Understandable Statistics: Concepts and Methods

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