91Ó°ÊÓ

The report "2005 Electronic Monitoring \& Surveillance Survey: Many Companies Monitoring, Recording, Videotaping-and Firing-Employees" (American Management Association, 2005 ) summarized the results of a survey of 526 U.S. businesses. Four hundred of these companies indicated that they monitor employees' web site visits. For purposes of this exercise, 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 \(.01 .\) 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 \(.01\).

Step by step solution

01

- Setting up the hypotheses

For both parts, a and b of the exercise, we will use the null hypothesis \(H_0\) and alternative hypothesis \(H_1\) to set up our hypotheses. For part a, we have to test the claim that more than 75% of businesses monitor employees' web visits. Thus, our null hypothesis would be \(H_0: p=0.75\) and alternative hypothesis would be \(H_1: p > 0.75\). For part b, we have to test the claim that more than 50% of businesses monitor employees' web visits. So, null hypothesis would be \(H_0: p=0.50\) and alternative hypothesis would be \(H_1: p > 0.50\)

02

- Calculate the test statistic

Use the formula for the test statistic, which is \(Z = (p_hat - p_0) / sqrt((p_0(1-p_0))/n)\) where \(p_hat\) is the sample proportion, \(p_0\) is the population proportion under the null hypothesis and \(n\) is the number of observations. For part a, \(p_hat\) is 400/526 = 0.76, \(p_0\) is 0.75, and \(n\) is 526. We now calculate and find the \(Z\) value. Then we will repeat this for part b with \(p_0\) as 0.50.

03

- Find the p-value

Next we find the p-value which is the probability of getting a test statistic as extreme as, or more extreme than, the observed test statistic, under the null hypothesis. We find this value by looking up our test statistic in a z-table or use a calculator.

04

- Conclusion

We compare our p-value to the given significance level of 0.01. If the p-value is less than or equal to 0.01, we reject the null hypothesis. If the p-value is more than 0.01, we fail to reject the null hypothesis. This will provide us with a conclusion about whether there is enough evidence to support the claim that more than 75% or more than 50% of businesses monitor 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

The null hypothesis is a statement used in statistics that proposes there is no significant difference or effect based on the data. Essentially, it's the default position that there is nothing unusual happening. In hypothesis testing, it's denoted by the symbol \(H_0\). For instance, if we're exploring whether a new teaching method is better than the current one, the null hypothesis would state there is no difference in effectiveness between the two methods. It acts as a starting point for statistical testing and is presumed true until evidence suggests otherwise.

Returning to our exercise, for part a, the null hypothesis (\(H_0: p = 0.75\)) asserts that 75% of U.S. businesses monitor employees’ web visits, which we accept as the truth until our analysis indicates a significant deviation.

Alternative Hypothesis

The alternative hypothesis, denoted as \(H_1\) or \(H_a\), is the statement that contradicts the null hypothesis by indicating there is a statistically significant effect or difference. It represents the outcome that the study is attempting to support. For example, in the context of the earlier example about teaching methods, the alternative hypothesis would state that the new teaching method has a different effectiveness than the traditional one.

In the exercise given, the alternative hypothesis for part a (\(H_1: p > 0.75\)) posits that more than 75% of businesses monitor, which explicitly suggests a specific direction of the difference—that the true proportion is greater than the declared 75%.

Significance Level

The significance level, often denoted by \(\alpha\), is a threshold set by the researcher to determine the cutoff for rejecting the null hypothesis. It's the probability of making a type I error, which is rejecting the null hypothesis when it is actually true. Common significance levels are 0.05, 0.01, and 0.10. The smaller the significance level, the more stringent the criterion for rejecting the null hypothesis, and the stronger the evidence must be.

In this exercise, the chosen significance level of 0.01 indicates we are allowing for a 1% risk of erroneously rejecting the null hypothesis. It is a rigorous standard, showing that we are requiring strong evidence before concluding that more than the stated percentage of businesses monitor their employees’ web visits.

P-value

The p-value is a crucial concept in hypothesis testing. It measures how compatible your data is with the null hypothesis. A low p-value suggests that your data is unlikely under the null hypothesis, which can lead you to reject the null hypothesis. The p-value answers the question: If the null hypothesis is true, what is the probability of observing a test statistic as extreme as, or more extreme than, what was observed?

In our exercise, the p-value signifies the likelihood of finding a sample proportion of businesses that monitor employee web usage as high as the one indicated by our survey data (or even higher), assuming that 75% (or 50%) is the actual percentage among all U.S. businesses. If this p-value is less than 0.01, the evidence is strong enough to challenge the null hypothesis.

Test Statistic

A test statistic is a standardized value derived from sample data during a hypothesis test. It's calculated by converting the observed effect (like a mean difference or, in this case, a proportion difference) into a number that measures how far it is from the expected effect under the null hypothesis. The test statistic allows us to determine how extreme the observed outcome is and hence, is pivotal in deciding whether or not to reject the null hypothesis.

For our example, the test statistic is found using the Z-formula, taking into account our sample proportion, the proportion under null hypothesis, and the sample size. This statistic is then used to calculate the p-value or can be directly compared to critical values from the standard normal distribution to draw conclusions.

Sample Proportion

Sample proportion is a statistic that indicates the fraction of the sample that has a certain characteristic, estimated as the number of 'successes' divided by the total sample size. It's symbolized as \(\hat{p}\). In the context of survey data or experiments, it allows us to make inferential statements about the population proportion.

In the case of the exercise, our sample proportion is the ratio of the number of companies that monitor employee web usage (the 'successes', which is 400) to the total number of companies surveyed (526). This gives us a sample proportion, \(\hat{p}\), of 0.76. It's this sample proportion that we compare to the hypothesized population proportion to conduct the hypothesis test.

Z-test

The Z-test is a statistical test used to determine whether two population means are different when the variances are known and the sample size is large. The Z-test is most appropriate when the sample size is at least 30. It uses the Z-distribution, which is the standard normal distribution, assuming that the effect size is large enough for the Central Limit Theorem to hold.

In the exercise, we're particularly dealing with a Z-test for a proportion, where we're comparing the sample proportion (\(\hat{p}\)) to a hypothesized proportion under the null hypothesis (\(p_0\)). The calculated Z value from the test statistic can then be used to determine the p-value, which in turn informs us whether to reject or not reject the null hypothesis based on our previously set significance level.