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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

The table below includes results from polygraph (lie detector) experiments conducted by researchers Charles R. Honts (Boise State University) and Gordon H. Barland (Department of Defense Polygraph Institute). In each case, it was known if the subject lied or did not lie, so the table indicates when the polygraph test was correct. Use a 0.05 significance level to test the claim that whether a subject lies is independent of the polygraph test indication. Do the results suggest that polygraphs are effective in distinguishing between truths and lies?

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Leading Digits

Benford's Law: Distributuon of leading digits

1

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2

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