Question

The National Automobile Dealers Association collects data on sales numbers, prices, and usage of new and used automobiles. One statistic of interest is the proportion of cars that are still on the road after 10 years. In a 2019 sample of 1500 on the road cars, 357 were 10 years old or older. Let
p
be the true proportion of on the road cars that are 10 years or older. Consider testing the hypotheses
H 0
​
:p=0.25
versus
H 1
​
:p<0.25
What is the P-value for your test? Round your answer to two decimal places.

Answer 1

The P-value for the hypothesis test is 0.1034 .Round to two decimal places, the P-value is 0.10

To find the P-value for this hypothesis test, we need to use the sample proportion of on the road cars that are 10 years or older, which is the number of cars in the sample that are 10 years or older divided by the total number of cars in the sample. In this case, the sample proportion is $\frac{357}{1500}= 0.238.$

We can then use this sample proportion to calculate the P-value, which is the probability of observing a sample proportion as extreme as the one we have seen, or even more extreme, given that the null hypothesis is true. In this case, the null hypothesis states that the true proportion of on the road cars that are 10 years or older is 0.25, so we need to calculate the probability of seeing a sample proportion as low as 0.238, or lower, if the true proportion is actually 0.25.

To do this, we can use a one-sided test, since we are only interested in the probability of seeing a sample proportion lower than 0.25. We can use a normal approximation to the binomial distribution to calculate the P-value.

The standard error of the sample proportion is given by the formula:

SE $= \sqrt\frac {p \ * (1 - p)}{n}$

Where p is the null hypothesis value for the proportion (0.25 in this case) and n is the sample size (1500 in this case). Plugging these values into the formula gives us:

SE = $\sqrt\frac{0.25 * (1 - 0.25)}{1500}= 0.0158$

The z-score for the sample proportion is then given by the formula:

z =$\frac{ (p - p_0)}{SE}$

Where p is the sample proportion (0.238 in this case) and p0 is the null hypothesis value for the proportion (0.25 in this case). Plugging these values into the formula gives us:

$z = \frac{(0.238 - 0.25)}{0.0158} = -1.27$

We can then use the z-score to calculate the P-value. Since we are performing a one-sided test, we only need to consider the probability of seeing a sample proportion lower than 0.25. This is given by the area under the normal curve to the left of the z-score. We can use a z-table or a calculator to find this probability.

The probability for a z-score of -1.27 is about 0.1034. This is the P-value for the hypothesis test.

Round to two decimal places, the P-value is 0.10

To learn more about probability, visit:

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