91Ó°ÊÓ

Many people who are involved with Major League Baseball believe that Yankees' baseball games tend to last longer than games played by other teams In order to test this theory, one other MLB team, the St. Louis Cardinals, was picked at random. The time of the game (in minutes) for 12 randomly selected Cardinals' games and 14 randomly selected Yankees' games was obtained. $$\begin{array}{cc} \text { Yankees } & \text { Cardinals } \\ \hline 155 & 208 \\\205 & 135 \\\190 & 161 \\\193 & 170 \\\232 & 150 \\\208 & 187 \\\174 & 200 \\\188 & 143 \\\229 & 154 \\\158 & 193 \\\202 & 128 \\\189 & 212 \\\232 & \\\211 & \\\\\hline\end{array}$$ Do these samples provide significant evidence to conclude that the mean time of Yankees' baseball games is significantly greater than the mean time of Cardinals' games? Use \(\alpha=0.05\).

Step by step solution

01

Calculate the Means

Calculate the mean time of games for each team. The mean is found by adding all of a set of values and then dividing by the number of values. Let's denote the mean time of Yankees' games as \( \mu_y \) and of Cardinals' games as \( \mu_c \).

02

Calculate the Standard Deviations

Calculate the sample standard deviation of game times for both teams. The standard deviation gives a measure of how spread out the numbers in a data set are. It is a square root of variance, which shows how much each number in a set differs from the mean. Let's denote the standard deviation of Yankees' games as \( \sigma_y \) and of Cardinals' games as \( \sigma_c \).

03

Calculate the t-Test Statistic

The formula for the t-statistic in an independent t-test is \(t = \frac{{\mu_y - \mu_c}}{{\sqrt{{\sigma_y^2/n_y + \sigma_c^2/n_c}}}}\), where \(n_y\) and \(n_c\) are the sample sizes of Yankees and Cardinals games respectively.

04

Compare t-Test Statistic with Critical Value

Compare the calculated t-statistic with the critical value from the t-distribution table. The critical value for a one-tailed test with \(\alpha=0.05\) and \(df=n_y+n_c-2\) degrees of freedom can be found in the table. If the calculated t-statistic is greater than the critical value, we reject the null hypothesis and conclude that there's significant evidence that mean time of Yankees' games is greater than Cardinals' games.

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91Ó°ÊÓ!

Key Concepts

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

t-test

The t-test is a statistical method used to determine if there is a significant difference between the means of two groups, which can be related in certain features. Here, we use it to compare the average game times of the Yankees and the Cardinals. The goal is to see whether the difference in their average (mean) game times is statistically meaningful. This is especially useful when the sample sizes are small, and the population variance is unknown. The t-test helps us decide whether to accept or reject the null hypothesis, which in this context states that the mean game times for the Yankees and the Cardinals are the same.

• The t-test calculates a "t-statistic," a value that helps determine the probability of observing the data if the null hypothesis were true.
• The greater the absolute value of the t-statistic, the more evidence there is against the null hypothesis.
• In this problem, the test is one-tailed since we are specifically checking if Yankees' game times are greater.

sample mean

The sample mean is the average of a sample and is used to estimate the true population mean. In the exercise about the MLB games, we calculate the sample means for Yankees and Cardinals to compare them. To find the sample mean, sum up all the recorded game times for each team and then divide by the number of games. This gives us the average game time across the sampled matches. It's important because it acts as a simple summary of game durations and provides a centerpiece for making comparisons with other data sets.

• For Yankees' games, sum all the times and divide by 14 (the number of Yankees games).
• For Cardinals' games, sum all the times and divide by 12 (the number of Cardinals games).
• The sample mean acts as the foundation for further statistical calculations, including the standard deviation and the t-test statistic.

standard deviation

Standard deviation is a measure of the amount of variation or dispersion in a set of values. A low standard deviation means that the values tend to be close to the mean of the set, while a high standard deviation means they are spread out over a wider range. In the context of the baseball game times, standard deviation lets us understand how consistent the game durations are for each team. Calculating the standard deviation involves several steps:

• First, subtract the sample mean from each individual game time to find the "deviation" from the mean.
• Then, square each deviation to eliminate negative values and emphasize larger deviations.
• Average these squared deviations to find the variance.
• Finally, take the square root of the variance to get the standard deviation. This measure is used in the t-test formula to assess the reliability of the sample mean.

critical value

A critical value is a point on the scale of the test statistic beyond which we reject the null hypothesis, and it corresponds to the significance level in hypothesis testing. It's essentially a threshold that determines the boundary of the rejection region for the test.In the example of Yankees and Cardinals game times, the critical value depends on the chosen significance level, \(\alpha=0.05\), and the degrees of freedom, which is calculated based on the sample sizes of both teams. You reference a t-distribution table to find the critical value.

• The critical value tells you how extreme the t-statistic must be to consider the results statistically significant.
• If the calculated t-statistic exceeds the critical value, the null hypothesis — that there is no difference in mean game times — is rejected.
• In this exercise, since the test is one-tailed, we only look for if the Yankees' game times statistically significantly exceed the Cardinals'.