Bundle: Introduction to Statistics and Data Analysis, 5th + WebAssign Printed Access Card: Peck/Olsen/Devore. 5th Edition, Single-Term
Bundle: Introduction to Statistics and Data Analysis, 5th + WebAssign Printed Access Card: Peck/Olsen/Devore. 5th Edition, Single-Term
5th Edition
ISBN: 9781305620711
Author: Roxy Peck, Chris Olsen, Jay L. Devore
Publisher: Cengage Learning
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Chapter 12.1, Problem 10E

a.

To determine

Test whether fatal bicycle accidents are equally likely to occur in each of the 12-months at 0.01 significance level.

a.

Expert Solution
Check Mark

Answer to Problem 10E

There is convincing evidence that fatal bicycle accidents are not equally likely to occur in each of the months.

Explanation of Solution

Calculation:

The given data represent the classification of fatal bicycle accidents according to the month in which the accident occurred.

The expected counts are calculated as shown below:

MonthObserved countsExpected counts
January3871912=59.917
February3271912=59.917
March4371912=59.917
April5971912=59.917
May7871912=59.917
June7471912=59.917
July9871912=59.917
August8571912=59.917
September6471912=59.917
October6671912=59.917
November4271912=59.917
December4071912=59.917
 719719

The nine-step hypotheses testing procedure to test goodness-of-fit is given below:

1. Consider that the proportion of fatal bicycle accidents occurring in January is p1, the proportion of fatal bicycle accidents occurring in February is p2, the proportion of fatal bicycle accidents occurring in March is p3, the proportion of fatal bicycle accidents occurring in April is p4, the proportion of fatal bicycle accidents occurring in May is p5, the proportion of fatal bicycle accidents occurring in June is p6, the proportion of fatal bicycle accidents occurring in July is p7, the proportion of fatal bicycle accidents occurring in August is p8, the proportion of fatal bicycle accidents occurring in September is p9, the proportion of fatal bicycle accidents occurring in October is p10, the proportion of fatal bicycle accidents occurring in November is p11, and the proportion of fatal bicycle accidents occurring in December is p12.

2. Null hypothesis:

H0:p1=p2=p3=p4=p5=p6=p7=p8=p9=p10=p11=p12=112.

3. Alternative hypothesis:

Ha: At least one of the population proportions is not equal to 112.

4. Significance level:

α=0.01

5. Test statistic:

χ2=(observed countexpected count)2expected count

6. Assumptions:

  • Assume that the 719 accidents included in the study is a random sample from the population of fatal bicycle accidents.
  • From the table, it is observed that all the expected counts are greater than 5.

7. Calculation:

Software procedure:

Step-by-step procedure to obtain the test statistics and P-value using the MINITAB software:

  • Choose Stat > Tables > Chi-Square Goodness-of-Fit Test (One Variable).
  • In Observed counts, enter the column of Number of Accidents.
  • In Category names, enter the column of Month.
  • Under Test, select the Equal Proportions.
  • Click OK.

The output obtained using the MINITAB software is given below:

Bundle: Introduction to Statistics and Data Analysis, 5th + WebAssign Printed Access Card: Peck/Olsen/Devore. 5th Edition, Single-Term, Chapter 12.1, Problem 10E , additional homework tip  1

From the output, χ2=82.1627.

8. P-value:

From the MINITAB output, df=11 and P-value is 0.000.

9. Conclusion:

Decision rule:

  • If P-value is less than or equal to the level of significance, reject the null hypothesis.
  • Otherwise, do not reject the null hypothesis.

Conclusion:

Here the level of significance is 0.01.

Here, P-value is less than the level of significance.

That is, 0.000<0.01.

Hence, reject the null hypothesis. Therefore, there is convincing evidence that fatal bicycle accidents are not equally likely to occur in each of the months.

b.

To determine

Write the null and alternative hypotheses to determine if some months are riskier than others by taking differing month lengths into account.

b.

Expert Solution
Check Mark

Explanation of Solution

The null hypothesis in Part (a) specifies that fatal accidents are equally likely to occur in any of the 12 months.

The year considered in the study is 2004, which is a leap year. It is known that a leap year contains 366 days, with 29 days in February.

The null and alternative hypotheses to determine if some months are riskier than others by taking differing month lengths and the characteristics of a leap year into account are as follows:

Null hypothesis:

H0:{p1=31366, p2=29366, p3=31366, p4=30366, p5=31366, p6=30366p7=31366, p8=31366, p9=30366, p10=31366, p11=30366,  p12=31366

Alternative hypothesis:

Ha: At least one of the population proportions is not equal to the specified value.

c.

To determine

Test the hypotheses proposed in Part (b) at 0.05 significance level.

c.

Expert Solution
Check Mark

Answer to Problem 10E

There is convincing evidence that fatal bicycle accidents do not occur in any of the twelve months in proportion to the lengths of the months.

Explanation of Solution

Calculation:

The expected counts are calculated as shown below:

MonthObserved counts

Proportion

pi (in decimal)

Expected counts
January380.08531366×719=60.90
February320.07929366×719=56.97
March430.08531366×719=60.90
April590.08230366×719=58.93
May780.08531366×719=60.90
June740.08230366×719=58.93
July980.08531366×719=60.90
August850.08531366×719=60.90
September640.08230366×719=58.93
October660.08531366×719=60.90
November420.08230366×719=58.93
December400.08531366×719=60.90
 7191 (approximately)719

Significance level:

α=0.05

Test statistic:

χ2=(observed countexpected count)2expected count

Assumptions:

  • Assume that the 719 accidents included in the study is a random sample from the population of fatal bicycle accidents.
  • From the table, it is observed that all the expected counts are greater than 5.

Calculation:

Software procedure:

Step-by-step procedure to obtain the test statistic and P-value using the MINITAB software:

  • Choose Stat > Tables > Chi-Square Goodness-of-Fit Test (One Variable).
  • In Observed counts, enter the column of Number of Accidents.
  • In Category names, enter the column of Month.
  • Under Test, select the column of Proportion in Proportions specified by historical counts.
  • Click OK.

The output obtained using the MINITAB software is given below:

Bundle: Introduction to Statistics and Data Analysis, 5th + WebAssign Printed Access Card: Peck/Olsen/Devore. 5th Edition, Single-Term, Chapter 12.1, Problem 10E , additional homework tip  2

From the output, χ2=78.1873.

8. P-value:

From the MINITAB output, df=11 and P-value is 0.000.

9. Conclusion:

Decision rule:

  • If P-value is less than or equal to the level of significance, reject the null hypothesis.
  • Otherwise, do not reject the null hypothesis.

Conclusion:

Here the level of significance is 0.05.

Here, P-value is less than the level of significance.

That is, 0.000<0.05.

Hence, reject the null hypothesis.

Therefore, there is convincing evidence that fatal bicycle accidents do not occur in any of the twelve months in proportion to the lengths of the months.

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Chapter 12 Solutions

Bundle: Introduction to Statistics and Data Analysis, 5th + WebAssign Printed Access Card: Peck/Olsen/Devore. 5th Edition, Single-Term

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