American League baseball teams play their games with the...

American League baseball teams play their games with the designated hitter rule, meaning that pitchers do not bat. The league believes that replacing the pitcher, traditionally a weak hitter, with another player in the batting order produces more runs and generates more interest among fans. Below are the average numbers of home runs hit per game in American League and National League stadiums for the 2013 season. a) Create an appropriate display of these data. What do you see? b) With a $95 \%$ confidence interval, estimate the mean number of home runs hit in American League games. c) Coors Field, in Denver, stands a mile above sea level, an altitude far greater than that of any other major league ball park. Some believe that the thinner air makes it harder for pitchers to throw curve balls and easier for batters to hit the ball a long way. Do you think the 1.96 home runs hit per game at Coors is unusual? (Denver is a National League team.) Explain. d) Explain why you should not use two separate confidence intervals to decide whether the two leagues differ in average number of runs scored. e) Using a $95 \%$ confidence interval, estimate the difference between the mean number of home runs hit in American and National League games. f) Interpret your interval. g) Does this interval suggest that the two leagues may differ in average number of home runs hit per game?

American League baseball teams play their games with the designated hitter rule, meaning that pitchers do not bat. The league believes that replacing the pitcher, traditionally a weak hitter, with another player in the batting order produces more runs and generates more interest among fans. Below are the average numbers of home runs hit per game in American League and National League stadiums for the 2013 season.
a) Create an appropriate display of these data. What do you see?
b) With a $95 \%$ confidence interval, estimate the mean number of home runs hit in American League games.
c) Coors Field, in Denver, stands a mile above sea level, an altitude far greater than that of any other major league ball park. Some believe that the thinner air makes it harder for pitchers to throw curve balls and easier for batters to hit the ball a long way. Do you think the 1.96 home runs hit per game at Coors is unusual? (Denver is a National League team.) Explain.
d) Explain why you should not use two separate confidence intervals to decide whether the two leagues differ in average number of runs scored.
e) Using a $95 \%$ confidence interval, estimate the difference between the mean number of home runs hit in American and National League games.
f) Interpret your interval.
g) Does this interval suggest that the two leagues may differ in average number of home runs hit per game?

Key Concepts

Pitfalls of Using Two Separate Confidence Intervals

Using two separate confidence intervals to compare groups can be misleading because overlapping confidence intervals do not necessarily imply the absence of a statistically significant difference, and vice versa. Instead, when comparing groups, it is more appropriate to construct a single confidence interval for the difference between means. This approach accounts for the joint variability of the estimates and provides a direct method to assess whether there is a statistically significant difference between the groups.

Comparison of Means

Comparison of means involves statistically evaluating whether the average values from different groups (in this case, American League and National League home runs per game) are significantly different. This is often achieved using methods such as constructing a confidence interval for the difference between means or performing hypothesis tests. Accurately comparing means is crucial for making valid inferences about differences between groups.

Outlier Analysis

Outlier analysis refers to identifying and assessing data points that deviate significantly from the overall pattern of the data. For example, the unusual performance at Coors Field might be examined to see if it represents an outlier due to its unique conditions. Understanding outliers is vital because they can influence statistical measures like the mean and may require special treatment or explanation in the analysis.

Confidence Interval

A confidence interval is a range of values, derived from sample data, which is likely to contain the true value of an unknown population parameter. In this context, a 95% confidence interval means that if we were to collect many samples and compute an interval from each sample, approximately 95% of those intervals would contain the true mean number of home runs per game. This concept is crucial in determining the precision of an estimate and providing a measure of uncertainty around the sample statistic.

Data Visualization

Data visualization involves creating graphical representations of data to identify patterns, trends, and outliers. In the context of comparing home run averages between leagues, appropriate displays such as box plots, histograms, or side-by-side bar charts help in visually assessing differences, central tendencies, and dispersions in the data sets. Effective visual displays are essential for initial data exploration and communicating findings clearly.

Transcript

00:02 First, to display the data, i would use something like decimus, if you wish, and you can just use their box plot feature.

00:09 You just enter the data.

00:11 You can call 1l and then use square brackets and then commas between the data.

00:19 And maybe call the other one n and do the same.

00:22 Enter the data.

00:24 And then use the box plot feature, which under statistics or histogram would be nice too.

00:29 And you can see the data.

00:31 You would see that the american league seems slightly higher on average and that the american league distribution seems to be more spread out than the national league.

00:43 So for the 95 % compensatable for the mean american league home runs, n is 15.

00:50 The mean is about 2.

00:56 So the american league standard deviation for a rough estimate using the formula for the standard deviation is about 0 .40.

01:04 If you needed to, you could calculate it more precisely.

01:13 The standard error is about 0 .40 over the square root of 15, and that's about 0 .103.

01:23 The critical t for 14 degrees of freedom at 95 % is 2 .145.

01:31 So the confidence interval is the mean plus or minus the critical t times are standard error, which is about 2 .00 plus or minus 0 .221.

01:52 So our confidence interval goes from 1 .78 to 2 .22.

01:58 So we're 95 % confident that the true mean home runs per game in the american league is between 1 .78 and 2 .22 home runs.

02:08 Is coorsfield unusual at 1 .96? at 1 .96 in a mean of 2, it's not unusual...

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