Question

A study reported that a certain percentage of cars registered in [year] had a manual transmission. An automotive researcher collected data on cars registered in [year] to investigate whether the proportion of registered cars with a manual transmission is less than [specified percentage]. The researcher took a random sample of cars registered in [year] and found [number] of the cars had a manual transmission. After ensuring the conditions for inference were met, the researcher conducted an appropriate hypothesis test and calculated a p-value of [p-value]. At the [percent] level of significance, which of the following conclusions should be made?

Responses:

a. The null hypothesis should be rejected. There is significant evidence that the population proportion of registered cars with a manual transmission in [year] is less than [specified percentage].

b. The null hypothesis should be rejected. There is significant evidence that the population proportion of registered cars with a manual transmission in [year] is less than 0.14.

c. The null hypothesis should be rejected. There is not significant evidence that the population proportion of registered cars with a manual transmission in [year] is less than [specified percentage].

d. The null hypothesis should be rejected. There is not significant evidence that the population proportion of registered cars with a manual transmission in [year] is less than 0.14.

e. The null hypothesis should not be rejected. There is significant evidence that the population proportion of registered cars with a manual transmission in [year] is not less than [specified percentage].

f. The null hypothesis should not be rejected. There is significant evidence that the population proportion of registered cars with a manual transmission in [year] is not less than 0.14.

g. The null hypothesis should not be rejected. There is significant evidence that the population proportion of registered cars with a manual transmission in [year] is equal to [specified percentage].

The null hypothesis should not be rejected. There is significant evidence that the population proportion of registered cars with a manual transmission in [year] is equal to 0.14.

The null hypothesis should not be rejected. There is not significant evidence that the population proportion of registered cars with a manual transmission in [year] is less than [specified percentage].

Answer 1

The decision whether to reject the null hypothesis is contingent on the comparison between the p-value and the alpha level. A p-value of 0.0417 against an alpha of 0.05 leads to rejection of the null hypothesis, providing evidence of a difference at the 5 percent significance level.

Step-by-step explanation:

At the specified level of significance, the conclusion depends on the comparison between the p-value and the alpha value. If the calculated p-value is less than the alpha (the threshold for significance), the null hypothesis, which is typically that there is no difference or no effect, should be rejected, suggesting that there is evidence to support the alternative hypothesis.

Given a p-value of 0.0417 and an alpha of 0.05, the decision is to reject the null hypothesis, as the p-value is lower than the alpha. This means that there is sufficient evidence at the 5 percent level of significance to conclude that the proportion is different from the null hypothesis value. However, if the alpha is set at a stricter level, such as 0.01, we would not reject the null hypothesis as the p-value is higher than the alpha, indicating insufficient evidence to conclude a difference at that significance level.