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The report "2007 Electronic Monitoring and Surveillance Survey: Many Companies Monitoring, Recording, Videotaping-and Firing-Employees" (American Management Association, 2007) summarized a survey of 304 U.S. businesses. Of these companies, 201 indicated that they monitor employees' web site visits. Assume that it is reasonable to regard this sample as representative of businesses in the United States. a. Is there sufficient evidence to conclude that more than \(75 \%\) of U.S. businesses monitor employees' web site visits? Test the appropriate hypotheses using a significance level of 0.01 . b. Is there sufficient evidence to conclude that a majority of U.S. businesses monitor employees' web site visits? Test the appropriate hypotheses using a significance level of 0.01 .

Step by step solution

01

Set Up the Hypotheses

In hypothesis testing, we always have two contradicting hypotheses: the null hypothesis (\(H_0\)) and the alternative hypothesis (\(H_a\)). For part (a) of our problem, we want to test if more than \(75\%\) of businesses control employees' website visits, so: \(H_0: p = 0.75\) and \(H_a: p > 0.75\). For part (b), we test if a majority of businesses, i.e., more than \(50\%\), control visits, so: \(H_0: p = 0.50\) and \(H_a: p > 0.50\). p refers to the proportion of all US businesses that monitor their employees' Internet usage.

02

Calculate Test Statistic and P-value

We determine the test statistic using the formula \(Z = (p - p_0) / \sqrt{(p_0 * (1 - p_0))/n}\). Where \(n\) is the sample size, \(p\) is the sample proportion, and \(p_0\) is the proportion under the null hypothesis. Using the sample data, for (a) we get \(Z1 = (201/304 - 0.75) / \sqrt{(0.75 * (1 - 0.75))/304} = -1.13\), and for (b), \(Z2 = (201/304 - 0.50) / \sqrt{(0.5 * (1 - 0.5))/304} = 10.26\). Then for each case, we calculate the p-value, which is the probability of getting a Z value as extreme as ours assuming the null hypothesis is true. p-value is given by \(P(Z > Z_{observed})\) for a right-tailed test.

03

Compare P-value to Significance Level

In each case, compare the calculated p-value to the provided significance level, \(\alpha\). If p-value is less than or equal to \(\alpha\), we reject the null hypothesis. If p-value is greater than \(\alpha\), we don't have enough evidence to reject the null hypothesis. Special tables (or computer software) give us the probability that Z is more than any given value.

04

Make Conclusion

Based on results from steps 2 and 3, make a conclusion for both parts. In part (a), as the p-value for Z1 would be larger than 0.01, we fail to reject the null hypothesis. We do not have enough evidence to support that more than \(75\%\) of U.S. businesses monitor employees' website visits. For part (b), since the p-value for Z2 will be less than 0.01, we reject the null hypothesis and conclude that the majority of U.S. businesses monitor their employees' web visits.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Null Hypothesis

When conducting a statistical test, the null hypothesis (\(H_0\)) represents a statement of no effect or no difference. In other words, it's the skeptical assumption that there is no significant association or outcome in the experiment or survey. For our exercise, the null hypothesis is developed for two separate cases:

• In part (a), we want to test if more than 75% of U.S. businesses monitor websites, so we assume initially that exactly 75% do, taking \(H_0: p = 0.75\).
• Meanwhile, in part (b), we're examining if more than 50% (a simple majority) monitor websites. Here, the null hypothesis assumes exactly 50% do, written as \(H_0: p = 0.50\).

It's important to define the null hypothesis clearly because it's the benchmark against which we measure our statistical findings. The null is either refuted or not refuted through our analysis.

P-value

The p-value is a key component in hypothesis testing. It tells us the probability of observing a test statistic as extreme as the one calculated, assuming the null hypothesis is true. It's the chance we would see our data, or something more extreme, purely from random chance if the null hypothesis were correct.
The p-value is used as a measure to make our decision. To determine if our data significantly differ from the hypothesis, we computed Z values, which lead us to our p-values:

• For part (a): With a Z value of -1.13, the p-value shows a probability that more than 75% of businesses do not monitor employees as intensely as thought. The exact probability helps us decide if our results are statistically significant by comparing it to a threshold known as the significance level.
• For part (b): Here, a Z value of 10.26 yields a much smaller p-value, indicating a strong probability that the true proportion of monitoring exceeds 50%, confirming our suspicion.

Remember, lower p-values indicate stronger evidence against the null hypothesis.

Significance Level

The significance level (\(\alpha\)) is a predetermined threshold used to judge the detected p-value. It often represents the risk level we are willing to take when rejecting a true null hypothesis. Commonly used \(\alpha\)values are 0.05, 0.01, or 0.10, but in our exercise, we use 0.01.
This represents a 1% risk level of concluding something significant when there is actually nothing significant.

• If the p-value is less than or equal to \(\alpha\), then we reject the null hypothesis, suggesting there's a statistically significant effect or difference.
• Conversely, if the p-value is greater than \(\alpha\), we fail to reject the null hypothesis, indicating our data isn't sufficiently surprising given the null hypothesis.

In our scenario, the outcomes were analyzed with a significance level of 0.01 to ensure robust insights.

Proportion

In the world of statistics, a proportion is a type of ratio representing a part of a whole. It's typically expressed as a fraction or percentage. In our context, proportions are crucial to understanding the extent of behavior or characteristic in a population, such as monitoring employees' web visits.
The formula for sample proportion (\(\hat{p}\)) is understood by dividing the number of successes (in this case, businesses that monitor) by the total number in the sample:

• With 201 companies out of 304 monitoring employees, the sample proportion calculates to \(\hat{p} = \frac{201}{304} \approx 0.6618\) or 66.18%.

The hypothesized population proportion (\(p_0\)) is what is assumed under the null hypothesis.

• For part (a), \(p_0 = 0.75\).
• For part (b), \(p_0 = 0.50\).

Understanding these proportions allows us to compare our observed data against what we would expect, thereby validating or challenging the null hypothesis.