A random sample of 25 ACME employees revealed that the average number of vacation days taken annually is 18.3, with a standard deviation of 3.72 days. Do these data provide sufficient evidence to suggest that ACME employees take more than 15 vacation days on average each year? (a) Appropriate hypotheses for this test: Null hypothesis (H0): μ = 15 (ACME employees take an average of 15 vacation days per year.) Alternative hypothesis (H1): μ > 15 (ACME employees take more than 15 vacation days on average each year.) (b) With a 5% level of significance, if the P-value for the test is 0.0392, we would reject the null hypothesis. This is because the P-value is less than the significance level (0.0392 < 0.05), indicating that there is sufficient evidence to support the claim that ACME employees take more than 15 vacation days on average each year. (c) A Type I error in the context of this test would occur if we incorrectly reject the null hypothesis when it is actually true. In this scenario, a Type I error means concluding that ACME employees take more than 15 vacation days on average when they do not. This would be an incorrect inference based on the sample data we have, likely due to random variation.

(a) The appropriate hypotheses for this test are:

Null hypothesis (H0): μ = 15 This is the hypothesis that ACME employees take an average of 15 vacation days per year, which is our baseline assumption.

Alternative hypothesis (H1): μ > 15 This is the hypothesis that ACME employees take more than 15 vacation days on average each year, which is what we’re testing for.

(b) With a 5% level of significance, if the P-value for the test is 0.0392, we would indeed reject the null hypothesis. This is because the P-value is below the significance level. In hypothesis testing, if the P-value is less than our chosen level of significance (in this case, 0.05), we have evidence to reject the null hypothesis in favor of the alternative hypothesis.

(c) A Type I error in the context of this test occurs if we reject the null hypothesis when in fact it is true. In other words, if ACME employees actually take an average of 15 vacation days per year, and we conclude based on our sample that they take more, we have made a Type I error. The probability of making a Type I error is equal to the significance level of the test, which in this case is 5%.

Extra:

Understand that when we perform a hypothesis test in statistics, we're trying to make inferences about a population based on sample data. The sample data is subject to random variability, so we must use statistical methods to determine whether any differences observed (like our sample mean of 18.3 days) are likely to be due to actual differences in the population or just due to chance.

The standard deviation is a measure of the variability or spread in a distribution of data. In our example, it is 3.72 days, which means that the vacation days for ACME employees are spread out by about 3.72 days from the average on either side.

A P-value is a probability that measures the evidence against the null hypothesis provided by the sample. A very small P-value suggests that it would be unlikely to get the sample data we have if the null hypothesis were true, which is why we might decide to reject it.

A Type I error, often denoted by alpha (α), is where we believe there is an effect or a difference when there isn't one -- a "false positive." This error is controlled by our choice of the significance level, which in most social sciences is set at 0.05, implying a 5% risk of committing a Type I error if the null hypothesis is true.

The level of significance is a threshold at which you decide how much uncertainty you are willing to accept. A significance level of 5% indicates you are willing to accept a 5% chance of incorrectly rejecting the true null hypothesis. This level can be thought of as a strictness measure; a lower significance level means you are being more strict with your evidence requirement to reject the null hypothesis.