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Chapter 7: Problem 68

The website www.mlb.com compiles statistics on all professional baseball players. For the 2017 season, statistics were recorded for all 663 players. Of this population, the mean batting average was \(0.236\) with a standard deviation of \(0.064\). Would it be appropriate to use this data to construct a \(95 \%\) confidence interval for the mean batting average of professional baseball players for the 2017 season? If so, construct the interval. If not, explain why it would be inappropriate to do so.

Short Answer

Expert verified
Yes, it is appropriate to construct a 95% confidence interval for this data. The interval is [0.2312, 0.2408]

Step by step solution

01

Verifying CLT conditions

To apply the Central Limit Theorem, normally three conditions need to be checked: Independence, Randomness, and Sample size (at least 30). In this case, we can assume these conditions are met. Each player's performance is independent of each other. The sample size of 663 is big enough (greater than 30). And we assume data are collected randomly to ensure that it's representative of the population.
02

Constructing 95% Confidence Interval

The formula for a confidence interval is \(\bar{x} 卤 Z_{\frac{1鈭�(1鈭扖onfidence\ level)/2}}*(\frac{蟽}{\鈭歂})\). For a 95% confidence interval, \(Z_{\frac{1鈭�(1鈭�0.95)/2}}=1.96\). The mean batting average \(\bar{x}=0.236\), the standard deviation 蟽=0.064, and the number of observations N=663. Now substitute these values into the formula: \(0.236 卤 1.96*(0.064/鈭�663)\).
03

Calculate the confidence interval

Let's calculate the right part of the formula: \(1.96*(0.064/鈭�663) = 0.0048\). Therefore, the 95% confidence interval for the mean batting average in 2017 is \(0.236 卤 0.0048 = [0.2312, 0.2408]\).

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Most popular questions from this chapter

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