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Baseball: Home Run Percentage The home run percentage is the number of home runs per 100 times at bat. A random sample of 43 professional baseball players gave the following data for home run percentages (Reference: The Baseball Encyclopedia, Macmillan). $$ \begin{array}{llllllllll} 1.6 & 2.4 & 1.2 & 6.6 & 2.3 & 0.0 & 1.8 & 2.5 & 6.5 & 1.8 \\ 2.7 & 2.0 & 1.9 & 1.3 & 2.7 & 1.7 & 1.3 & 2.1 & 2.8 & 1.4 \\ 3.8 & 2.1 & 3.4 & 1.3 & 1.5 & 2.9 & 2.6 & 0.0 & 4.1 & 2.9 \\ 1.9 & 2.4 & 0.0 & 1.8 & 3.1 & 3.8 & 3.2 & 1.6 & 4.2 & 0.0 \\ 1.2 & 1.8 & 2.4 & & & & & & & \end{array} $$ (a) Use a calculator with mean and standard deviation keys to verify that \(\bar{x} \approx 2.29\) and \(s \approx 1.40\). (b) Compute a \(90 \%\) confidence interval for the population mean \(\mu\) of home run percentages for all professional baseball players. Hint: If you use Table 6 of Appendix II, be sure to use the closest \(d . f\). that is smaller. (c) Compute a \(99 \%\) confidence interval for the population mean \(\mu\) of home run percentages for all professional baseball players. (d) Interpretation The home run percentages for three professional players are Tim Huelett, \(2.5 \quad\) Herb Hunter, \(2.0 \quad\) Jackie Jensen, \(3.8\) Examine your confidence intervals and describe how the home run percentages for these players compare to the population average. (e) Check Requirements In previous problems, we assumed the \(x\) distribution was normal or approximately normal. Do we need to make such an assumption in this problem? Why or why not? Hint: See the central limit theorem in Section \(6.5\).

Step by step solution

01

Verify the Mean and Standard Deviation

Calculate the mean \( \bar{x} \) and standard deviation \( s \) using a statistical calculator or software. Given the data, ensure \( \bar{x} \approx 2.29 \) and \( s \approx 1.40 \) are accurate results.

02

Calculate the 90% Confidence Interval

Use the formula for the confidence interval \( \bar{x} \pm t \, \frac{s}{\sqrt{n}} \), where \( t \) is the t-value from the t-distribution table for \( 90\% \) confidence and \( n-1 \) degrees of freedom. With \( n = 43 \), determine \( t \approx 1.682 \) and compute the interval.

03

Calculate the 99% Confidence Interval

Use the same confidence interval formula: \( \bar{x} \pm t \, \frac{s}{\sqrt{n}} \). Find the \( t \)-value for \( 99\% \) confidence with \( 42 \) degrees of freedom, approximately \( t \approx 2.423 \), and substitute into the formula.

04

Interpret Player Statistics

Compare the home run percentages of Tim Huelett (2.5), Herb Hunter (2.0), and Jackie Jensen (3.8) to the calculated confidence intervals. See if these percentages fall within the intervals to determine if they are typical or unusual.

05

Evaluate Assumptions

Based on the central limit theorem, as the sample size is \( 43 \), which is considered large enough, the sampling distribution of \( \bar{x} \) is approximately normal regardless of the distribution of the data.

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Key Concepts

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

Confidence Intervals

Confidence intervals give us a range of values that likely contain the population mean. They are essential in statistics, providing a measure of uncertainty regarding our sample estimates. When constructing a confidence interval, we use the sample mean, which represents our best estimate for the population mean. However, since we're drawing from a sample, we also need to account for variability. This is where the standard deviation and the t-distribution (or sometimes the z-distribution) come into play.

To calculate a confidence interval:

• Identify your desired confidence level (e.g., 90% or 99%).
• Find the corresponding t-value from the t-distribution table based on your sample size, minus one (degrees of freedom).
• Use the formula: \[ \bar{x} \pm t \frac{s}{\sqrt{n}} \]where \( \bar{x} \) is the sample mean, \( s \) is the standard deviation, and \( n \) is the sample size.

This results in a range where the actual population mean is likely to be found. For example, for a 90% confidence interval, if your interval is 2.2 to 2.4, this means you are 90% confident the true mean falls within this range.

Confidence intervals provide a way to quantify uncertainty, helping to make informed decisions based on statistical data.

Central Limit Theorem

The Central Limit Theorem (CLT) is one of the most important ideas in statistics. It states that the distribution of sample means will be approximately normally distributed, regardless of the shape of the population distribution, as long as the sample size is sufficiently large. This is good news for statisticians because it justifies the use of normal distribution methods in many situations.

Key Points of the Central Limit Theorem:

• As the sample size increases, the sampling distribution of the sample mean becomes more normally distributed.
• This theorem holds true no matter what the shape of the population distribution is – it could be skewed or have outliers.
• For most practical purposes, a sample size of 30 is considered large enough to apply the CLT.

Because of the CLT, we can apply methods that assume a normal distribution, like calculating confidence intervals, even if our data doesn't come from a normal distribution, provided our sample size is large enough. In the context of the baseball player statistics, because we have 43 data points, we can be confident that the sample mean follows a normal distribution, aiding in our confidence interval calculations.

Sample Mean

The sample mean, denoted as \( \bar{x} \), is the average of the data values in a sample and serves as an estimate for the population mean. In our baseball example, the sample mean of home run percentages is calculated by adding together all the percentages and dividing by the number of players in the sample (43, in this case).

Formula to calculate the sample mean:\[ \bar{x} = \frac{\sum_{i=1}^{n} x_i}{n} \]where \( x_i \) are the data points and \( n \) is the number of observations in the sample.
The sample mean is a crucial component in constructing confidence intervals and conducting hypothesis tests. It's important to remember that the sample mean is an estimation; sometimes, it might differ from the true population mean, which is why confidence intervals are useful. They provide a range based on this sample estimate that we're fairly confident includes the population mean.

Standard Deviation

Standard deviation is a measure of variability or dispersion in a dataset. It quantifies how much the data points deviate from the mean. A low standard deviation means that data points are close to the mean, while a high standard deviation indicates that the data points are spread out over a wider range.

In the formula for the confidence interval, \[ \bar{x} \pm t \frac{s}{\sqrt{n}} \],standard deviation \( s \) plays a key role in determining the "width" of the interval. The larger the standard deviation, the wider the interval, reflecting more uncertainty about the population mean.

Calculating the standard deviation involves:

• Finding the mean of your dataset.
• Subtracting the mean from each number to find the deviation for each value.
• Squaring these deviations, summing them up, and dividing by \( n - 1 \) to find the variance.
• The standard deviation is the square root of this variance.

This statistic helps in understanding the spread of data and is critical when making statistical inferences, including constructing confidence intervals and testing hypotheses.