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A statistics student believes that black cars are less likely to receive tickets for moving violations. Black cars make up 19% of all cars manufactured. The student randomly selects 70 moving violation records and finds that 10 of them involved black cars. The [tex]$P$[/tex]-value for the test of the hypotheses, [tex]$H_0: P=0.19$[/tex] and [tex]$H_a: P\ \textless \ 0.19$[/tex], is 0.24.

What is the correct interpretation of this value?

A. There is a 24% chance of a black car receiving a moving violation.

B. Assuming the true proportion of black cars that receive moving violations is 0.19, there is a 24% probability that the null hypothesis is true.

C. Assuming the true proportion of black cars that receive moving violations is 0.19, there is a 24% probability that the sample proportion would be 0.15 or less by chance alone.

D. Assuming the true proportion of black cars that receive moving violations is less than 0.19, there is a 24% probability that the sample proportion would be 0.15 or less by chance alone.



Answer :

GinnyAnswer
To interpret the given [tex]$P$[/tex]-value of 0.24 for the test of the hypothesis, we need to understand what a [tex]$P$[/tex]-value means in the context of hypothesis testing.

Step-by-Step Solution:

1. Formulate the Hypotheses:
- Null Hypothesis ([tex]$H_0$[/tex]): The true proportion of black cars that receive moving violations is 0.19.
- Alternative Hypothesis ([tex]$H_9$[/tex]): The true proportion of black cars that receive moving violations is less than 0.19.

2. Understand the P-value:
- The [tex]$P$[/tex]-value is the probability of observing a sample statistic (in this case, the sample proportion of 10 out of 70 black cars with violations, which is approximately 0.143) as extreme as, or more extreme than, the one observed, assuming that the null hypothesis is true.

3. Given the P-value:
- The [tex]$P$[/tex]-value is 0.24. This means that there is a 24% probability of obtaining a sample proportion of 0.15 or less (since 10/70 is approximately 0.143), or one that is even more extreme, assuming that the true proportion of black cars that receive moving violations is 0.19.

4. Interpret the P-value:
- The correct interpretation of the [tex]$P$[/tex]-value in this context is the probability of observing the sample result if the null hypothesis ([tex]$H_0$[/tex]) were true. Specifically, it tells us about the likelihood of obtaining a sample proportion as extreme as 0.143, given that the true proportion is 0.19.

Among the given options, the correct interpretation is:
- Assuming the true proportion of black cars that receive moving violations is 0.19, there is a 24% probability that the sample proportion would be 0.15 or less by chance alone.

Therefore, the correct interpretation is:
Assuming the true proportion of black cars that receive moving violations is 0.19, there is a 24% probability that the sample proportion would be 0.15 or less by chance alone.

This matches the correct interpretation choice:
Assuming the true proportion of black cars that receive moving violations is 0.19, there is a 24% probability that the sample proportion would be 0.15 or less by chance alone.

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