Question

Testing Claims About Proportions. In Exercises 7–22, test the given claim. Identify the null hypothesis, alternative hypothesis, test statistic, P-value or critical value(s), then state the conclusion about the null hypothesis, as well as the final conclusion that addresses the original claim.

License Plate Laws The Chapter Problem involved passenger cars in Connecticut and passenger cars in New York, but here we consider passenger cars and commercial trucks. Among2049 Connecticut passenger cars, 239 had only rear license plates. Among 334 Connecticuttrucks, 45 had only rear license plates (based on samples collected by the author). A reasonable hypothesis is that passenger car owners violate license plate laws at a higher rate than owners of commercial trucks. Use a 0.05 significance level to test that hypothesis.

a. Test the claim using a hypothesis test.

b. Test the claim by constructing an appropriate confidence interval.

Short answer

a. The hypotheses:

\(\begin{array}{l}{H_0}:{p_1} = {p_2}\\{H_1}:{p_1} > {p_2}\end{array}\)

The test statistic is -0.95 and critical value is 1.645.

Fail to reject the null hypothesis.

There is sufficient evidence to reject the claim thatpassenger car owners violate license plate laws at a higher rate than owners of commercial trucks.

b. The 90% confidence interval under is\( - 0.0510 < {p_1} - {p_2} < 0.0148\).There is sufficient evidence to reject the claim that passenger car owners violate license plate laws at a higher rate than owners of commercial trucks.

Step-by-step solution

Given information

In testing of owners who violate license plate laws, among 2049 Connecticut passenger’s cars only 239 cars had license plates, and among 334 Connecticut trucks, only 45 had license plates.

Level of significance to test hypothesis is 0.05.

Describe the hypothesis to be tested.

a. Let\({p_1}\)be the population proportion of car owners who violate rear license plate laws and\({p_2}\)be the population proportion of truck owners who violate rear license plate laws.

Mathematically, the test hypothesis is

\(\begin{array}{l}{H_0}:{p_1} = {p_2}\\{H_1}:{p_1} > {p_2}\end{array}\)

Thus, the test is right tailed.

Describe the formula of test staistics.

Test statistics is given by,

\(z = \frac{{\left( {{{\hat p}_1} - {{\hat p}_2}} \right) - \left( {{p_1} - {p_2}} \right)}}{{\sqrt {\frac{{\bar p\bar q}}{{{n_1}}} + \frac{{\bar p\bar q}}{{{n_2}}}} }}\)

Where,\({\hat p_{1{\rm{ }}}}and{\rm{ }}{\hat p_{{\rm{2 }}}}\)are estimated proportions,\(\bar p{\rm{ }}\)is pooled proportions and\(\bar q = 1 - \bar p\).

Also, \({n_{1{\rm{ }}}}and{\rm{ }}{n_2}\) are sample size of for each of the sample.

Calculate estimated values of proportions

Let \({\hat p_1}\)be the estimated proportion of car owners who violate license plate laws and\({\hat p_2}\)be the estimated proportion of truck owners who violates rear license plate laws.

Thus,

\(\begin{array}{c}{{\hat p}_1} = \frac{{239}}{{2049}}\\{{\hat p}_1} = 0.1166\\{{\hat q}_1} = 1 - {{\hat p}_1}\\{{\hat q}_1} = 0.8834\end{array}\)

Also,

\(\begin{array}{c}{{\hat p}_2} = \frac{{45}}{{334}}\\{{\hat p}_2} = 0.1347\\{{\hat q}_2} = 1 - {{\hat p}_2}\\{{\hat q}_{}} = 0.8653\end{array}\)

Here,\({x_1}\)and\({x_2}\)are number of owners who violate rear license plate laws of cars and trucks respectively.

The pooled proportions is computed as,

\(\begin{array}{c}\bar p = \frac{{{x_1} + {x_2}}}{{{n_1} + {n_2}}}\\ = \frac{{239 + 45}}{{2049 + 334}}\\ = 0.1192\\\bar q = 0.8808\end{array}\)

Thus, \(\bar p\bar q = 0.1050\)

Compute the value of test statistic

Test statistic is given by,

\(\begin{array}{c}z = \frac{{\left( {{{\hat p}_1} - {{\hat p}_2}} \right) - \left( {{p_1} - {p_2}} \right)}}{{\sqrt {\frac{{\bar p\bar q}}{{{n_1}}} + \frac{{\bar p\bar q}}{{{n_2}}}} }}\\ = \frac{{\left( {0.1166 - 0.1347} \right)}}{{\sqrt {\frac{{0.1050}}{{2049}} + \frac{{0.1050}}{{334}}} }}\\ = - 0.9461\end{array}\)

Thus, the test statistic is -0.95.

Conclude on the basis of critical value

If test statistic is greater than critical value, reject null hypothesis under\({\rm{\alpha }}\)level of significance.

Refer to z-table for the critical value. The right-tailed test implies the cumulative area to the left of the critical value would be 0.95(1-0.05).

Thus, the critical value is 1.645.

Here, the value of the test statistic is less than the critical value.

Therefore, we failed to reject the null hypothesisunder a 0.05 significance level.

There is sufficient evidence to reject the claim that passenger car owners violate license plate laws at a higher rate than owners of commercial trucks.

Describe the confidence interval.

b. The formula for confidence interval to test the claim is about difference between two population proportion\({p_1}{\rm{ and }}{p_2}\)is given by,

\(\left( {{{\hat p}_1} - {{\hat p}_2}} \right) - E < {p_1} - {p_2} < \left( {{{\hat p}_1} - {{\hat p}_2}} \right) + E\)

Where, E is margin of error.

Calculate margin of error

The 0.05 level of significance for one-tailed test implies 90% level of confidence.

The critical value for 90% confidence level is obtained as 1.645\(\left( {{z_{\frac{{0.10}}{2}}}} \right)\) using the standard normal table.

E is margin of error which is given by,

\(\begin{array}{c}E = {z_{\frac{\alpha }{2}}}\sqrt {\frac{{{{\hat p}_1}{{\hat q}_1}}}{{{n_1}}} + \frac{{{{\hat p}_2}{{\hat q}_2}}}{{{n_2}}}} \\ = 1.645 \times \sqrt {\frac{{0.1028}}{{2049}} + \frac{{0.1165}}{{334}}} \\ = 0.0329\end{array}\)

Construct the confidence interval

Substitute the valuesfor 90% confidence interval as,

\(\begin{array}{l}\left( {{{\hat p}_1} - {{\hat p}_2}} \right) - E < {p_1} - {p_2} < \left( {{{\hat p}_1} - {{\hat p}_2}} \right) + E = (0.1164 - 0.1347) - 0.033 < {p_1} - {p_2} < (0.1164 - 0.1347) + 0.033\\\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; = - 0.0510 < {p_1} - {p_2} < 0.0148\end{array}\)

The 90% confidence interval under is\( - 0.0510 < {p_1} - {p_2} < 0.0148\).

This interval contains zero that means there is no significant difference between two proportions\({{\rm{p}}_{\rm{1}}}{\rm{ and }}{{\rm{p}}_{\rm{2}}}\).

Therefore, the confidence interval suggests that there is no sufficient evidence to support the claim that passenger car owners violate license plate laws at a higher rate than owners of commercial trucks.