Introduction to Probability and Statistics
Introduction to Probability and Statistics
14th Edition
ISBN: 9781133103752
Author: Mendenhall, William
Publisher: Cengage Learning
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Chapter 12.4, Problem 12.13E

(a)

To determine

To find: the independent variable and the dependent variable.

(a)

Expert Solution
Check Mark

Answer to Problem 12.13E

The variable API is the response variable (y) or the dependent variable and the variable ELL is the predictor variable (x) or independent variable.

Explanation of Solution

Given:

    School

      1

      2

      3

      4

      5

      6

      7

      8

    API745808798791854688801751
    EL7118245017711157

Calculation:

From the given data,

The variable API is the response variable (y) or the dependent variable and the variable ELL is the predictor variable (x) or independent variable. Because, the Academic performance index (API) depends upon English language learner (ELL). Based on English language learner (ELL), we can measure Academic performance index (API).

Conclusion: The variable API is the response variable (y) or the dependent variable and the variable ELL is the predictor variable (x) or independent variable.

(b)

To determine

To plot:the given data using the scatterplot and explain whether the relationship between x and y is reasonable.

(b)

Expert Solution
Check Mark

Answer to Problem 12.13E

The assumptions of a linear relationship between x and y are reasonable.

Explanation of Solution

Given:

    School

      1

      2

      3

      4

      5

      6

      7

      8

    API745808798791854688801751
    EL7118245017711157

Calculation:

  Introduction to Probability and Statistics, Chapter 12.4, Problem 12.13E , additional homework tip  1

The assumptions of a linear relationship between x and y are reasonable because there is no strong curvature present in the scatterplot.

Conclusion: Therefore, the assumptions of a linear relationship between x and y are reasonable.

(c)

To determine

To calculate: the least-squares regression line.

(c)

Expert Solution
Check Mark

Answer to Problem 12.13E

The least squares regression line is y^=847.4766561.70453x .

Explanation of Solution

Given:

    School

      1

      2

      3

      4

      5

      6

      7

      8

    API745808798791854688801751
    EL7118245017711157

Calculation:

Calculation for the data in first two columns of the table is:

    API(y)ELL(x)

      x2

      y2

      xy

    74571504155502552895
    8081832465286414544
    7982457663680419152
    79150250062568139550
    8541728972931614518
    68871504147334448848
    801111216416018811
    75157324956400142807
    y=6,236x=319x2=17,141y2=4,878,636xy=241,125

The sum of squares are,

  Sxx=xi2( x i ) 2 n=17141 (319)28=4420.88

  Sxy=xiyi x i y i n=241125319×62368=7535.5

  Syy=yi2( y i ) 2 n=4878636 (6236)28=17674

The slope is,

  b=S xyS xx=60601993.5=1.70453

The mean is the sum of all values divided by the number of values.

  x¯= x i n=3198=39.875

  y¯= y i n=62368=779.5

The estimate a of the intercept is,

  a=y¯bx¯=779.50(1.70453)×39.88=847.476656

The least squares regression line is:

  y^=a+bx=847.4766561.70453x

Conclusion: Thus, the least squares regression line is y^=847.4766561.70453x .

(d)

To determine

To plot:the line on the scatterplot in part b and explain whether the line fits the data points.

(d)

Expert Solution
Check Mark

Answer to Problem 12.13E

The line fits through the data points.

Explanation of Solution

Given:

    School

      1

      2

      3

      4

      5

      6

      7

      8

    API745808798791854688801751
    EL7118245017711157

Calculation:

We plot the line on the scatter plot in the part (b):

  Introduction to Probability and Statistics, Chapter 12.4, Problem 12.13E , additional homework tip  2

The line fits through the data points. Since, there is equal deviation from the fitted line.

Conclusion: The line fits through the data points.

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Introduction to Probability and Statistics

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