Question

You have developed a series of questions to measure job satisfaction of bus drivers in New York City. A random sample of 100 drivers has an average score of \(10.6\), with a standard deviation of \(2.8\). What is your estimate of the average job satisfaction score for the population as a whole? Use the \(95 \%\) confidence level.

Short answer

The 95% confidence interval for the population mean is .

Step-by-step solution

Identify Given Information

Extract the given information from the problem. Here, the sample size () is 100, the sample mean () is 10.6, and the sample standard deviation () is 2.8. We need to estimate the population mean with a 95% confidence level.

Determine the Z-Score for 95% Confidence Interval

For a 95% confidence interval, the Z-score is 1.96. This is derived from standard normal distribution tables.

Calculate the Standard Error of the Mean (SEM)

The Standard Error of the Mean (SEM) is calculated using the formula: .. Plug in the numbers to get:.

Compute the Margin of Error (MOE)

The Margin of Error (MOE) can be calculated as: MOE = Z . Substituting the Z-score and SEM, we get: .

Calculate the Confidence Interval

The confidence interval is calculated by subtracting and adding the MOE from the sample mean. Thus, , giving the interval: .

Key concepts

Sample Mean

The sample mean is a fundamental concept in statistics. It represents the average score or value of a set of observations. In our exercise, the sample mean of the job satisfaction scores for NYC bus drivers is 10.6. To find the sample mean, you sum up all the individual scores and then divide by the number of observations. For example, if you had five scores: 12, 10, 11, 9, and 11, the sample mean would be calculated as:

\( \frac{12 + 10 + 11 + 9 + 11}{5} = 10.6 \).

This sample mean is used as an estimate for the population mean, which is what we aim to determine with a confidence interval.

Standard Deviation

Standard deviation measures the amount of variation or dispersion in a set of values. Simply put, it tells us how much the individual scores deviate from the sample mean. In our example, the standard deviation of the job satisfaction scores is 2.8. To calculate the standard deviation, follow these steps:

• Find the mean (average) of the data set.
• Subtract the mean from each score to get the deviation of each score.
• Square each deviation.
• Find the mean of these squared deviations.
• Take the square root of this mean to get the standard deviation.

The formula for standard deviation (s) is: \( s = \sqrt{ \frac{ \sum (X_i - \overline{X})^2 }{n-1} } \), where \( X_i \) represents each individual score, \( \overline{X} \) is the sample mean, and \( n \) is the number of observations.

Z-Score

The Z-score is a statistical measure that describes the number of standard deviations a data point is from the mean. It is used in the context of confidence intervals to determine how far away from the mean we want our interval to extend. For a 95% confidence interval, the Z-score is 1.96. This value comes from standard normal distribution tables and essentially tells us the range within which we can be 95% confident the true population mean lies. To calculate a Z-score, you can use the formula:

\( Z = \frac{X - \overline{X}}{s} \), where \(X\) is the value of interest, \( \overline{X} \) is the sample mean, and \( s \) is the standard deviation.

Standard Error of the Mean (SEM)

The Standard Error of the Mean (SEM) estimates the variability between sample means that you would obtain if you took multiple samples from the same population. It is calculated using the formula:

\( SEM = \frac{s}{\sqrt{n}} \), where \( s \) is the standard deviation and \( n \) is the sample size.

In our example, with a sample size of 100 and a standard deviation of 2.8, the SEM is:

\( \frac{2.8}{\sqrt{100}} = 0.28 \).

The smaller the SEM, the more precise the estimate of the population mean.

Margin of Error (MOE)

The Margin of Error (MOE) represents the range within which we can expect the true population mean to fall, given a specific confidence level. It is calculated by multiplying the Z-score by the SEM:

\( MOE = Z \times SEM \).

Using our earlier values, the MOE for a 95% confidence interval is:

\( 1.96 \times 0.28 = 0.5488 \).

This means our point estimate of 10.6 could reasonably fall within 0.5488 points above or below the mean. Therefore, the 95% confidence interval is:

\( 10.6 \pm 0.5488 \), or from \( 10.0512 \) to \( 11.1488 \).

This offers a range in which the true job satisfaction mean for all NYC bus drivers likely lies, with 95% confidence.