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In a study of high school students at least 16 years of age, researchers obtained survey results summarized in the accompanying table (based on data from 鈥淭exting While Driving and Other Risky Motor Vehicle Behaviors Among U.S. High School Students,鈥 by O鈥橫alley, Shults, and Eaton, Pediatrics,Vol. 131, No. 6). Use a 0.05 significance level to

test the claim of independence between texting while driving and driving when drinking alcohol. Are those two risky behaviors independent of each other?


Drove when drinking Alcohol?


Yes

No

Texted while driving

731

3054

No Texting while driving

156

4564

Short Answer

Expert verified

Texting while driving and driving while drunk are dependent behaviours of driving.

Step by step solution

01

Given information

The data forthe counts of subjects whotext while driving and who drive while drunk is provided.

The level of significance is 0.05.

02

Compute the expected frequencies

Theformula forexpected frequencyis,

\(E = \frac{{\left( {row\;total} \right)\left( {column\;total} \right)}}{{\left( {grand\;total} \right)}}\)

The table for observed frequencies with row and column total is represented as,


Drove when drinking Alcohol?



Yes

No

Row total

Texted while driving

731

3054

3785

No Texting while driving

156

4564

4720

Column total

887

7618

8505

The table for expected frequency is represented as,


Drove when drinking Alcohol?


Yes

No

Texted while driving

394.744

3390.256

No Texting while driving

492.256

4227.744

Each of the expected value is greater than 5. It is assumed that the subjects are selected randomly.

03

State the null and alternate hypothesis

\({H_0}:\)Texting while driving and driving while drunk are independent.

\({H_1}:\)Texting while driving and driving while drunk are dependent.

04

Compute the test statistic

The value of the test statisticis computed as,

\[\begin{aligned}{c}{\chi ^2} = \sum {\frac{{{{\left( {O - E} \right)}^2}}}{E}} \\ = \frac{{{{\left( {731 - 394.744} \right)}^2}}}{{394.744}} + \frac{{{{\left( {3054 - 3390.256} \right)}^2}}}{{3390.256}} + ... + \frac{{{{\left( {4564 - 4227.744} \right)}^2}}}{{4227.744}}\\ = 576.224\end{aligned}\]

Therefore, the value of the test statistic is 576.224.

05

Compute the degrees of freedom

The degrees of freedomare computed as,

\(\begin{aligned}{c}\left( {r - 1} \right)\left( {c - 1} \right) = \left( {2 - 1} \right)\left( {2 - 1} \right)\\ = 1\end{aligned}\)

Therefore, the degrees of freedom are 1.

06

Compute the critical value

From chi-square table, the critical value for row corresponding to 1 degree of freedom at 0.05 level of significance is 3.841.

Therefore, the critical value is 3.841.

Also, the p-value is computed as 0.000.

07

State the decision

Since the critical (3.841) is less than the value of test statistic (576.224). In this case, the null hypothesis is rejected.

Therefore, the decision is to reject the null hypothesis.

08

State the conclusion

There is not enough evidenceto support the claim that the two behaviours; textingwhile driving and driving while drunk with alcohol are independent.

Thus, it can be concluded that two risky behaviours are dependent on each other.

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Most popular questions from this chapter

Exercises 1鈥5 refer to the sample data in the following table, which summarizes the last digits of the heights (cm) of 300 randomly selected subjects (from Data Set 1 鈥淏ody Data鈥 in Appendix B). Assume that we want to use a 0.05 significance level to test the claim that the data are from a population having the property that the last digits are all equally likely.

Last Digit

0

1

2

3

4

5

6

7

8

9

Frequency

30

35

24

25

35

36

37

27

27

24

When testing the claim in Exercise 1, what are the observed and expected frequencies for the last digit of 7?

Benford鈥檚 Law. According to Benford鈥檚 law, a variety of different data sets include numbers with leading (first) digits that follow the distribution shown in the table below. In Exercises 21鈥24, test for goodness-of-fit with the distribution described by Benford鈥檚 law.

Leading Digits

Benford's Law: Distributuon of leading digits

1

30.10%

2

17.60%

3

12.50%

4

9.70%

5

7.90%

6

6.70%

7

5.80%

8

5.10%

9

4.60%

Detecting Fraud When working for the Brooklyn district attorney, investigator Robert Burton analyzed the leading digits of the amounts from 784 checks issued by seven suspect companies. The frequencies were found to be 0, 15, 0, 76, 479, 183, 8, 23, and 0, and those digits correspond to the leading digits of 1, 2, 3, 4, 5, 6, 7, 8, and 9, respectively. If the observed frequencies are substantially different from the frequencies expected with Benford鈥檚 law, the check amounts appear to result from fraud. Use a 0.01 significance level to test for goodness-of-fit with Benford鈥檚 law. Does it appear that the checks are the result of fraud?

Mendelian Genetics Experiments are conducted with hybrids of two types of peas. If the offspring follow Mendel鈥檚 theory of inheritance, the seeds that are produced are yellow smooth, green smooth, yellow wrinkled, and green wrinkled, and they should occur in the ratio of 9:3:3:1, respectively. An experiment is designed to test Mendel鈥檚 theory, with the result that the offspring seeds consist of 307 that are yellow smooth, 77 that are green smooth, 98 that are yellow wrinkled, and 18 that are green wrinkled. Use a 0.05 significance level to test the claim that the results contradict Mendel鈥檚 theory.

Exercises 1鈥5 refer to the sample data in the following table, which summarizes the last digits of the heights (cm) of 300 randomly selected subjects (from Data Set 1 鈥淏ody Data鈥 in Appendix B). Assume that we want to use a 0.05 significance level to test the claim that the data are from a population having the property that the last digits are all equally likely.

Last Digit

0

1

2

3

4

5

6

7

8

9

Frequency

30

35

24

25

35

36

37

27

27

24

If using a 0.05 significance level to test the stated claim, find the number of degrees of freedom.

In Exercises 5鈥20, conduct the hypothesis test and provide the test statistic and the P-value and , or critical value, and state the conclusion.

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