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. The report stated that 137 of the 526 businesses had fired workers for misuse of the Internet and 131 had fired workers for e-mail misuse. For purposes of this exercise, assume that it is reasonable to regard this sample as representative of businesses in the Unired Srates. a. Construct and interpret a \(95 \%\) confidence interval for the proportion of U.S. businesses that have fired workers for misuse of the Internet. b. What are two reasons why a \(90 \%\) confidence interval for the proportion of U.S. businesses that have fired workers for misuse of e-mail would be narrower than the \(95 \%\) confidence interval computed in Part (a)?

Step by step solution

01

Identify the Proportions and Sample Size

The proportion who have fired for internet misuse, \(p_{internet}\), is 137 out of 526, i.e., \(p_{internet} = 137 / 526\). Similarly, the proportion for email misuse, \(p_{email}\), is 131 out of 526, i.e., \(p_{email} = 131 / 526\). The sample size, n, is 526.

02

Calculate Standard Errors

The standard error of proportion, \(SE_{p_{internet}}\), is calculated as \(\sqrt{(p_{internet} * (1 - p_{internet}) / n)}\). Similarly, the standard error for email misuse, \(SE_{p_{email}}\), is calculated using the same formula.

03

Construct 95% Confidence Interval for Internet Misuse

A 95% confidence interval is calculated as \(p_{internet} \pm z * SE_{p_{internet}}\), where z is the z-score for a 95% confidence interval, which is approximately 1.96.

04

Explain Narrower 90% Confidence Interval for Email Misuse

The 90% confidence interval for the proportion of businesses that have fired workers for email misuse would be narrower than the 95% confidence interval computed in part (a) because of two reasons: a) The level of confidence is less, meaning that we are willing to accept a higher chance of error, hence the interval does not need to be as wide. b) The z-score for a 90% confidence interval would be lower (approximately 1.645) than that of a 95% confidence interval, leading to a decrease in the overall margin of error and hence, a narrower interval. These factors contribute to a smaller range, while still considering the same sample data.

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Key Concepts

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

Proportion

In statistics, the term *proportion* refers to a part or fraction of a total. It is usually expressed as a decimal or percentage and represents the size of a certain group relative to the whole. In the exercise, two proportions are calculated:

• The proportion of businesses that have fired employees for internet misuse: to find this, we divide the number of businesses that fired employees (137) by the total number of businesses surveyed (526). This gives us a proportion of approximately 0.260. This means about 26% of businesses reported firing for internet misuse.
• The proportion for email misuse, calculated as 131 divided by 526, is about 0.249 or 24.9%.

Proportions are essential when making estimates about a larger population since they give us insight into how common a feature or behavior is within that group.

Sample Size

The term *sample size* (often denoted by 'n') refers to the number of observations or data points that are collected in a study. In any statistical analysis, the sample size influences the accuracy and reliability of the results. Larger sample sizes tend to give more reliable results because they better approximate the true population.
In this exercise, the sample size is 526, meaning that data was collected from 526 businesses. This relatively large sample size helps to ensure that the proportions calculated for those fired for internet and email misuse are reflective of the broader population of U.S. businesses. It minimizes the potential effects of randomness and sampling error, providing a solid basis for generalizing findings to a wider group.

Standard Error

The *standard error* measures the variability or spread of a statistic, like a proportion, from the true population parameter. It informs us about how much we can expect a sample statistic to vary due to random sampling error. Standard error is crucial for determining confidence intervals.
To calculate the standard error of a proportion, we use the formula:\[SE = \sqrt{\frac{p(1-p)}{n}}\]Here, *p* represents the sample proportion, and *n* is the sample size. This equation gives us insight into how precise our sample proportion is and allows us to construct a confidence interval that will estimate the true population proportion.

Z-score

A *z-score* is a statistical measurement that describes a value's position within a distribution relative to the mean. It is a key component in the creation of confidence intervals.
The z-score tells us how many standard deviations away a particular measurement is from the mean. In confidence intervals, a z-score helps to determine the probability that a statistic falls within a certain range.

• For example, a 95% confidence interval uses a z-score of about 1.96, which indicates that 95% of the data will fall within approximately 1.96 standard deviations of the mean if the data is normally distributed.
• For a 90% confidence interval, the z-score is lower, approximately 1.645, resulting in a narrower interval.

By using the z-score along with the standard error, statisticians can accurately construct confidence intervals to capture the true population parameter within a specified level of confidence.