Problem 40 - Basic Statistics for Business and Economics with Student

Question

During recent seasons, Major League Baseball has been criticized for the length of the games. A report indicated that the average game lasts 3 hours and 30 minutes. A sample of 17 games revealed the following times to completion. (Note that the minutes have been changed to fractions of hours, so that a game that lasted 2 hours and 24 minutes is reported at 2.40 hours.) Can we conclude that the mean time for a game is less than 3.50 hours? Use the .05 significance level. $$\begin{array}{|lllllllll|}\hline 2.98 & 2.40 & 2.70 & 2.25 & 3.23 & 3.17 & 2.93 & 3.18 & 2.80 \\\2.38 & 3.75 & 3.20 & 3.27 & 2.52 & 2.58 & 4.45 & 2.45 & \\\\\hline\end{array}$$

Step-by-Step Solution

Answer

The mean game time is less than 3.50 hours at the 0.05 significance level.

Step 1: Define the Hypotheses

We need to define the null and alternative hypotheses. The null hypothesis (H0) states that the mean time for a game is equal to or greater than 3.50 hours. The alternative hypothesis (H1) states that the mean time for a game is less than 3.50 hours. In mathematical terms, this is:- Null Hypothesis: \( H_0: \mu \geq 3.50 \)- Alternative Hypothesis: \( H_1: \mu < 3.50 \)

Step 2: Calculate the Sample Mean

The next step is to calculate the mean of the sample data. Add all the game times and divide by the number of games (17) to find the average. The sample data is: \(2.98, 2.40, 2.70, 2.25, 3.23, 3.17, 2.93, 3.18, 2.80, 2.38, 3.75, 3.20, 3.27, 2.52, 2.58, 4.45, 2.45\). The mean \(\bar{x}\) is calculated as follows:\[ \bar{x} = \frac{2.98 + 2.40 + 2.70 + 2.25 + 3.23 + 3.17 + 2.93 + 3.18 + 2.80 + 2.38 + 3.75 + 3.20 + 3.27 + 2.52 + 2.58 + 4.45 + 2.45}{17} \approx 3.00 \]

Step 3: Calculate the Sample Standard Deviation

Calculate the standard deviation of the sample. The formula for the sample standard deviation \( s \) is:\[ s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n-1}} \]Substitute the values to compute \( s \).

Step 4: Determine the Test Statistic

Using the sample mean \( \bar{x} \), the standard deviation \( s \), and the number of observations \( n \), calculate the t-statistic. The formula for the t-statistic is:\[ t = \frac{\bar{x} - \mu_0}{s/\sqrt{n}} \] where \( \mu_0 = 3.50 \) is the population mean under the null hypothesis.

Step 5: Find the Critical Value

For a one-tailed test at a 0.05 significance level and \( n-1 = 16 \) degrees of freedom, use the t-distribution table to find the critical value. In this case, the critical value \( t_{critical} \approx -1.746 \).

Step 6: Make a Decision

Compare the calculated t-statistic with the critical value. If the t-statistic is less than the critical value, reject the null hypothesis. Otherwise, do not reject the null hypothesis.

Step 7: Result Interpretation

If we reject the null hypothesis, it suggests that the mean time for a game is indeed less than 3.50 hours at the 0.05 significance level.

Key Concepts

Null and Alternative Hypotheses

When conducting a hypothesis test, the first step is to establish the null and the alternative hypotheses. These are foundational for any statistical test and guide the analysis.

• Null Hypothesis (\(H_0\)): This hypothesis represents the default assumption. In this context, the null hypothesis claims that the mean time of Major League Baseball games is equal to or greater than 3.50 hours. Mathematically, we express this as \( H_0: \mu \geq 3.50 \).
• Alternative Hypothesis (\(H_1\)): This is what you want to prove through your test. Here, the alternative hypothesis proposes that the mean time is less than 3.50 hours. Its mathematical notation is \( H_1: \mu < 3.50 \).

In hypothesis testing, we either reject the null hypothesis in favor of the alternative or fail to reject it. If enough evidence supports that the mean time is indeed less than 3.50 hours, we reject \(H_0\) and accept \(H_1\). Otherwise, we conclude that our data does not provide enough evidence to support a decrease in mean time.

Sample Mean Calculation

Calculating the sample mean is an essential step in hypothesis testing, giving us the average of our collected data. Think of the mean as the center point of all the numbers in your set.

Here's how you calculate it:

• Add all the sample data points together. In our case: 2.98, 2.40, 2.70, etc. Add these values to get the total sum.
• Divide the total sum by the number of observations. There are 17 games, so you divide the total sum by 17.
• The formula: \( \bar{x} = \frac{\sum x_i}{n} \)

This simple arithmetic computation yields the sample mean, which is approximately 3.00 hours for this data set. The sample mean serves as an estimate of the population mean and is crucial for further statistical analysis.

T-Statistic

The t-statistic is a value you calculate to decide whether to reject the null hypothesis. It compares the difference between your sample mean and the population mean (hypothesized mean) in units of standard error.

Here’s how you calculate the t-statistic:

• Calculate the sample standard deviation \( s \). This measures how much the game times vary from the average game time in your sample. The formula for standard deviation is \( s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n-1}} \).
• Compute the standard error \( \frac{s}{\sqrt{n}} \). It's the standard deviation of your sample mean and represents how much your sample mean would vary if you took multiple samples.
• Use the formula for the t-statistic: \( t = \frac{\bar{x} - \mu_0}{s/\sqrt{n}} \); where \( \mu_0 \) is the hypothesized population mean (3.50 hours here).

The t-statistic tells us how far your sample mean is from the hypothesized mean, in standard error units. If this value is significantly low (less than the critical value from the t-distribution table), it suggests that the observed sample mean is rare under the null hypothesis, leading us to reject it.