Question

In addition to teachers and administrative staff, schools also have many other employees, including bus drivers, custodians, and cafeteria workers. The average hourly wage is \(\$ 14.18\) for bus drivers, \(\$ 12.61\) for custodians, and \(\$ 10.33\) for cafeteria workers. \(^{21}\) Suppose that a school district employs \(n=36\) bus drivers who earn an average of \(\$ 11.45\) per hour with a standard deviation of \(s=\) \$2.84. Find a \(95 \%\) confidence interval for the average hourly wage of bus drivers in school districts similar to this one. Does your confidence interval enclose the stated average of \(\$ 14.18 ?\) What can you conclude about the hourly wages for bus drivers in this school district?

Short answer

Answer: Yes, the average hourly wage for bus drivers in this school district may be lower than the stated average for similar school districts, as the 95% confidence interval does not include the stated average hourly wage of $14.18.

Step-by-step solution

Determine the degrees of freedom

The degrees of freedom for a t-distribution with sample size \(n\) is given by \(d f=n-1\). For our sample, we have \(d f=36-1=35\).

Calculate the t-value corresponding to the confidence level

We are asked for a 95% confidence interval. Therefore, we need to find the t-value (\(t_{\alpha/2}\)) such that 95% of the area under the t-distribution (with 35 degrees of freedom) is enclosed. From a t-table or calculator, we can find that \(t_{\alpha/2} = 2.030\).

Calculate the margin of error

The margin of error for a 95% confidence interval using the t-distribution is given by \(E = t_{\alpha/2} \times \frac{s}{\sqrt{n}}\), where \(s = \$2.84\) and \(n = 36\). Plugging in the values, we get \(E = 2.030 \times \frac{\$2.84}{\sqrt{36}} =\$0.97\).

Construct the confidence interval

The 95% confidence interval for the population mean is given by \(\bar{x} \pm E = \$11.45 \pm \$0.97\), which results in the interval \((\$11.45 - \$0.97, \$11.45 + \$0.97) = (\$10.48, \$12.42)\).

Compare the confidence interval to the stated average

Our 95% confidence interval for the average hourly wage of bus drivers in similar school districts is \((\$10.48, \$12.42)\). The stated average hourly wage, \(\$ 14.18\), is not inside the confidence interval.

Conclusion

Since the stated average hourly wage of \(\$ 14.18\) is not within our 95% confidence interval for the population mean, we can conclude that the hourly wages for bus drivers in this school district may be lower than the stated average for similar school districts.

Key concepts

T-Distribution

The t-distribution, also known as the Student's t-distribution, is a probability distribution that is symmetric and bell-shaped, similar to the normal distribution but with heavier tails. It arises when estimating the mean of a normally distributed population in situations where the sample size is small and the population standard deviation is unknown.

The t-distribution is broader and more prone to producing values that fall far from the mean, which reflects the increased uncertainty that comes with smaller samples. As the sample size grows, the t-distribution approaches the normal distribution. This is why it's particularly important when dealing with small sample sizes, as in the exercise where there are only 36 bus drivers.

Degrees of Freedom

Degrees of freedom is a concept tied closely to statistical estimations and the t-distribution. In the context of the given problem, it refers to the number of values in the calculations of a statistic that are free to vary. To calculate degrees of freedom for a t-distribution, one typically subtracts one from the sample size, that is, \(n-1\).

In the given exercise, for example, with a sample size of 36 bus drivers, the degrees of freedom is 35. This value is crucial as it determines the exact shape of the t-distribution that is used to estimate the confidence interval for the average hourly wage.

Margin of Error

The margin of error reflects the extent to which the sample mean can vary from the true population mean. It's a way to express the confidence in the statistical inference. Calculating the margin of error involves the t-distribution, particularly the relevant t-value for the chosen confidence level, the sample standard deviation, and the sample size.

For our exercise, it represents the range either side of the sample mean within which we expect the true mean to lie, given a certain level of confidence. The margin of error was calculated using the formula \( E = t_{\alpha/2} \times \frac{s}{\sqrt{n}} \) and it integrates the concept of t-distribution with the degrees of freedom and the sample size to arrive at a precise measure.

Population Mean Estimation

Estimating the population mean involves inferring the central value of a population parameter based on a random sample from the population. In the situation of the bus drivers, we use the sample mean and the margin of error to estimate the population mean, being aware that the true mean hourly wage could be any value within the confidence interval.

The calculated interval from the exercise \( (\$10.48, \$12.42) \) suggests that, with 95% confidence, the true average hourly wage for all bus drivers in similar school districts lies in this range. This estimation process underlines the importance of incorporating margin of error and understanding the limitations of sample-based inferences.

Statistical Inference

Statistical inference encompasses the procedures of drawing conclusions about a population's characteristics based on a sample. It is through inference that we use sample data to estimate population parameters like means or proportions. Confidence intervals are a key part of statistical inference, providing a range of values which is likely to include the parameter of interest.

In our example, by constructing a 95% confidence interval we're asserting that there’s a 95% chance that the interval \( (\$10.48, \$12.42) \) contains the true average hourly wage. This interval is based on the sample mean and provides valuable insight, even though the population mean, which is unknown, might lie outside of this range. The exercise illustrates how statistical inference allows us to make educated guesses about the population characteristics from a sample.