Problem 33 - Basic Statistics for Business & Economics

Question

The National Collegiate Athletic Association (NCAA) reported that college football assistant coaches spend a mean of 70 hours per week on coaching and recruiting during the season. A random sample of 50 assistant coaches showed the sample mean to be 68.6 hours, with a standard deviation of 8.2 hours. a. Using the sample data, construct a \(99 \%\) confidence interval for the population mean. b. Does the \(99 \%\) confidence interval include the value suggested by the NCAA? Interpret this result. c. Suppose you decided to switch from a \(99 \%\) to a \(95 \%\) confidence interval. Without performing any calculations, will the interval increase, decrease, or stay the same? Which of the values in the formula will change?

Step-by-Step Solution

Answer

a. \\( [65.494, 71.706] \\) b. Yes, it includes 70, indicating plausibility. c. The interval will decrease; the critical value changes.

Step 1: Gather required data

First, identify the given data for constructing the confidence interval. We have the sample mean \( \bar{x} = 68.6 \) hours, sample standard deviation \( s = 8.2 \) hours, and sample size \( n = 50 \). We need to calculate a 99% confidence interval for the population mean.

Step 2: Determine the critical value

For a 99% confidence interval and a sample size of 50, we use the t-distribution because the population standard deviation is unknown. The degrees of freedom will be \( n - 1 = 49 \). From the t-table, the critical value \( t^* \) for a 99% confidence level and 49 degrees of freedom is approximately 2.680.

Step 3: Calculate the standard error of the mean

Calculate the standard error \( SE \) using the formula:\[SE = \frac{s}{\sqrt{n}}\]\[SE = \frac{8.2}{\sqrt{50}} \approx 1.159\]

Step 4: Construct the confidence interval

Calculate the confidence interval using the formula:\[\bar{x} \pm t^* \times SE\]\[68.6 \pm 2.680 \times 1.159\]\[68.6 \pm 3.106\]This results in a confidence interval of \([65.494, 71.706]\) hours.

Step 5: Compare with NCAA reported mean

Check if the population mean reported by NCAA (70 hours) is within the calculated interval. The interval \([65.494, 71.706]\) includes the value 70, suggesting that the NCAA's reported mean is plausible within our calculated interval.

Step 6: Discuss confidence interval change without calculation

If we switch from a 99% to a 95% confidence interval, the interval will decrease because we're accepting a lower confidence level, which uses a smaller critical value. The t-critical value decreases, resulting in a narrower interval.

Key Concepts

Sample Mean

The sample mean is a critical component in statistical analysis, serving as the average value of a sample data set. In our exercise, the sample mean is given as 68.6 hours per week.
Using the sample mean, we can infer about the population mean, which is usually unknown, by creating confidence intervals.
Finding the sample mean involves adding all data points in a sample and dividing by the number of those points.
It provides us with a number that represents the central tendency of our sample:

• Helps in estimating population parameters.
• Forms the basis for calculating the confidence interval.

It is important to note the distinction between the sample mean and population mean. The latter refers to the average of the entire population, which is usually not feasible to calculate.

T-Distribution

When we want to estimate a confidence interval for a population mean, especially with unknown population standard deviation, we rely on the t-distribution. Unlike the normal distribution, the t-distribution accounts for small sample sizes by being wider and flatter.
The t-distribution becomes more like the normal distribution as sample size increases.
In our exercise, we used a sample of 50 coaches which leads us to use the t-distribution rather than the normal distribution.

• Used if population standard deviation is unknown.
• Accounts for variability when sample size is small.
• Approached with degrees of freedom, which is calculated as sample size minus one ( - 1).

This is crucial, especially in constructing confidence intervals, to provide more conservative and likely accurate intervals when dealing with limited data.

Standard Error

The standard error (SE) quantifies how much the sample mean is expected to vary from the true population mean. It is the standard deviation of the sample mean's distribution and is calculated by dividing the sample standard deviation by the square root of the sample size:
\[ SE = \frac{s}{\sqrt{n}} \]The smaller the standard error, the more reliable the sample mean as an estimate of the population mean.
SE is specifically vital when constructing confidence intervals, as it adjusts the width of such an interval.

• Reflects the accuracy of the sample mean.
• A smaller SE indicates that the sample mean is a better estimate of the population mean.
• Changes inversely with sample size – larger the sample, smaller the SE.

In our task, the SE was calculated to be approximately 1.159, signifying how reliable the sample mean of 68.6 hours per week is.

Critical Value

The critical value in the context of confidence intervals represents the cutoff point for the confidence level we choose. It affects the range of the confidence interval and depends on the specified confidence level and the underlying distribution:

• Higher confidence levels mean wider intervals due to higher critical values.
• Critical value is extracted from the t-distribution table using confidence level and degrees of freedom.
• For our exercise with a 99% confidence level, the critical value was approximately 2.680.

Switching from a 99% confidence level to a 95% confidence level reduces the critical value, which in turn narrows the confidence interval. It's a trade-off between interval width and level of certainty.