Question

InasurveyconductedbyCareerBuilders.com,employers were asked if they had ever sent an employee home because he or she was dressed inappropriately (June \(17,2008,\) www .careerbuilders.com). A total of 2,765 employers responded to the survey, with 968 saying that they had sent an employee home for inappropriate attire. In a press release, CareerBuilder makes the claim that more than one-third of employers have sent an employee home to change clothes. Do the sample data provide convincing evidence in support of this claim? Test the relevant hypotheses using \(\alpha=\) 0.05 . For purposes of this exercise, assume that the sample is representative of employers in the United States.

Short answer

To come to a decision whether or not the sample data support CareerBuilder's claim that more than one-third of employers have sent an employee home to change clothes, we need to carry out a hypothesis test. If the p-value is less than the significance level \(\alpha = 0.05\), then the hypothesis is supported.

Step-by-step solution

Understand the Data and the Hypothesis

Using the data provided, we know that a sample of 2,765 employers was asked if they sent an employee home for inappropriate attire, with 968 employers affirming this. Our assumption is that more than one-third of employers have done so.

Develop the Hypotheses

The Null Hypothesis \((H_0)\) is that the population proportion \(p\) is 1/3 or 0.33, which suggests that only one third of all employers have sent an employee home for inappropriate attire. The Alternative Hypothesis \((H_1)\) is that the population proportion \(p\) is greater than 1/3 or 0.33, indicating more than one-third of employers have asked an employee to go back home for inappropriately dressing.

Calculate the Test Statistic

Using the normal approximation to the binomial distribution, we can calculate the z-score using the formula mentioned above: \[Z = \rac{\(p̂ - p_0\)}{\sqrt{\rac{\(p_0(1 - p_0\)}{n}}}\], where, \(p̂\) is the sample proportion (which is 968/2765 or 0.35), \(p_0\) is the proportion under \(H_0\) which is 0.33, and \(n\) is the sample size which is 2765.

Find the p-value

After calculating the z-score, we next need to find the p-value, which is the probability of getting a test statistic as extreme as the calculated z-score, given that \(H_0\) is true. This would be a one-sided test as we are only interested in whether the proportion is greater than 0.33.

Conclusion

After obtaining the p-value, if it's less than \(\alpha=\) 0.05, we will reject the null hypothesis in favor of the alternative. This would lend support to CareerBuilder's claim. If not, we do not have enough evidence to support the claim.

Key concepts

Proportion Hypothesis Test

When we conduct a proportion hypothesis test, we're essentially examining whether the proportion of a certain characteristic within a population differs from a specific value. In the given exercise, CareerBuilders.com wants to ascertain if more than one-third of employers have sent an employee home for inappropriate attire based on a sample survey.

To test this claim statistically, we compare the observed proportion in the sample (which is the number of employers who said they sent an employee home divided by the total number of surveyed employers) to the claimed proportion (one-third). If the observed proportion significantly differs from the claimed proportion, then we have evidence to support or refute the survey's claim. It’s critical that our sample is representative of the broader population, as the validity of the hypothesis test is dependent on this representativeness.

Null Hypothesis

The null hypothesis, often denoted as \(H_0\), represents a statement of no effect or no difference; it's what we assume to be true before collecting any data. In our example, the null hypothesis posits that the true proportion of employers (\(p\)) that have sent employees home for inappropriate attire is exactly one-third, or 0.33.

The null hypothesis is a starting point for statistical tests, offering a baseline measure against which the alternative hypothesis is compared. We never prove the null hypothesis; rather, we look for evidence to reject it in favor of the alternative hypothesis. If we don't find such evidence, we simply fail to reject the null, cautiously maintaining the status quo regarding our belief.

Alternative Hypothesis

The alternative hypothesis, notated as \(H_1\) or \(H_a\), is what we aim to support with our data. This hypothesis is a statement that suggests a new observation or a difference from the status quo. For the CareerBuilders.com survey, the alternative hypothesis claims that the true proportion of employers who have sent employees home (\(p\)) is greater than one-third, supporting the statement made in the press release.

If our data provides sufficient evidence against \(H_0\), we can then reject the null hypothesis in favor of this alternative scenario. The alternative hypothesis is crucial because it represents the substantive claim or association we're interested in demonstrating with our data.

P-Value

The p-value is a probability that measures the strength of the evidence against the null hypothesis provided by our sample. It tells us how likely it is to observe our sample data, or something even more extreme, assuming the null hypothesis is true. The smaller the p-value, the stronger the evidence against \(H_0\).

In the exercise, after calculating the test statistic (a z-score), we determine the p-value associated with it. This p-value will show us the likelihood of observing a sample proportion of employers as large as we have (or larger) if the true proportion is truly one-third. A p-value less than the chosen significance level \(\alpha = 0.05\) means that the data is sufficiently unlikely under the null hypothesis that we reject \(H_0\), thus supporting the alternative hypothesis.

Statistical Significance

The concept of statistical significance is foundational in hypothesis testing. When our results are 'statistically significant', it means that what we've observed in our sample is unlikely to be due to just random chance - according to the threshold we set with our significance level, often \(\alpha\). This level is a tolerance for making an error in rejecting a true null hypothesis, usually set at 0.05, or 5%.

If the p-value is below this \(\alpha\) threshold, we declare our results as statistically significant, rejecting the null hypothesis. In our CareerBuilder example, if the p-value calculated from the test statistic is below 0.05, we would conclude that there is statistically significant evidence to support the claim that more than one-third of employers have sent an employee home for inappropriate attire - hence the importance of understanding this critical threshold in hypothesis testing.