Question

California, Connecticut, and New York are states with laws requiring that cars have license plates on the front and rear. The author randomly selected cars in those states, and the results are given in the accompanying table. Use a 0.05 significance level to test the claim of independence between the state and whether a car has front and rear license plates. Does it appear that the license plate laws are followed at the same rates in the three states?

	California	Connecticut	New York
Car with Rear Plate Only	35	45	9
Car with Front and Rear Plates	528	289	541

Short answer

The state and whether a car has front and rear license plates are dependent.

Also, the rules for execution of license plate rules are not followed at the same rates in three states.

Step-by-step solution

Given information

The data for the states and the position of car plates is given.

Compute the expected frequencies

Compute theexpected frequencyusing the row and columns totals of the observed values,

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

The table for observed values and the respective row and column total is represented as,

	California	Connecticut	New York	Row total
Car with Rear Plate Only	35	45	9	89
Car with Front and Rear Plates	528	289	541	1358
Column total	563	334	550	1447

Theexpected frequency tableis represented as,

	California	Connecticut	New York
Car with Rear Plate Only	34.628	20.543	33.829
Car with Front and Rear Plates	528.372	313.457	516.171

Assume that each subject is randomly selected along with the fact that each expected frequency is greater than 5.

Then, the requirements for the test are satisfied.

State the null and alternate hypothesis

The hypotheses are formulated as,

\({H_0}:\)The state and whether a car has front and rear license plates are independent.

\({H_1}:\)The state and whether a car has front and rear license plates are dependent.

Compute the test statistic

The value of the test statisticis computed as,

\(\begin{aligned}{c}{\chi ^2} = \sum {\frac{{{{\left( {O - E} \right)}^2}}}{E}} \\ = \frac{{{{\left( {35 - 34.628} \right)}^2}}}{{34.628}} + \frac{{{{\left( {45 - 20.543} \right)}^2}}}{{20.543}} + ... + \frac{{{{\left( {541 - 516.171} \right)}^2}}}{{516.171}}\\ = 50.4458\end{aligned}\)

Therefore, the value of the test statistic is 50.4458.

Compute the degrees of freedom

The degrees of freedomare computed as,

\(\begin{aligned}{c}\left( {r - 1} \right)\left( {c - 1} \right) = \left( {2 - 1} \right)\left( {3 - 1} \right)\\ = 2\end{aligned}\)

Therefore, the degrees of freedom are 2.

Compute the critical value

From the chi-square table, the critical value corresponding to 2 degrees of freedom and at 0.05 level of significance 5.992.

Therefore, the critical value is 5.992.

The p-value is obtained as 0.000.

State the decision

Since the critical value (5.992) is less than the value of the test statistic (50.4458). In this case, the null hypothesis is rejected.

Therefore, the decision is to reject the null hypothesis.

State the conclusion

There is insufficient evidence to support the claimthat the state and whether a car has front and rear license plates are independent.

Thus, the execution of rules for license plates is dependent on the states.

The proportion of cars in the study that followed the law of license plate, both front and rear of the car.

	California	Connecticut	New York
Cars with front and rear plates	0.9378	0.8653	0.9836

Of the three states, all appear to follow the rule by different rates.