91Ó°ÊÓ

In a study comparing age of death for left- and right-handed baseball players, Coren and Halpern (1991, p. 93) provided the following information: "Mean age of death for strong right-handers was 64.64 years (SD = 15.5, n = 1472); mean age of death for strong left-handers [was] 63.97 years ( SD \(=15.4, n=236\) )" The term "strong handers" applies to baseball players who both threw and batted with the same hand. The data were actually taken from entries in The Baseball Encyclopedia (6th ed., New York: Macmillan, 1985), but, for the purposes of this exercise, pretend that the data were from a sample drawn from a larger population. a. Compute an approximate \(95 \%\) confidence interval for the mean age of death for the population of strong right-handers from which this sample was drawn. b. Repeat part (a) for the strong left-handers. c. Compare the results from parts (a) and (b) in two ways. First, explain why one confidence interval is substantially wider than the other. Second, explain whether you would conclude that there is a difference in the mean ages of death for left-and right-handers on the basis of these results. d. Compute an approximate \(95 \%\) confidence interval for the difference in mean ages of death for the strong right-and left-handers. Interpret the result.

Step by step solution

01

Calculate the 95% Confidence Interval for Right-Handers

To calculate the 95% confidence interval for the mean age of death of strong right-handers, use the formula for the confidence interval: \[\text{CI} = \bar{x} \pm z \left(\frac{s}{\sqrt{n}}\right)\]where \(\bar{x} = 64.64\) (mean), \(s = 15.5\) (standard deviation), \(n = 1472\) (sample size), and \(z = 1.96\) for a 95% confidence level. Calculating the margin of error (ME):\[\text{ME} = 1.96 \times \left(\frac{15.5}{\sqrt{1472}}\right) \approx 0.793\]Thus, the confidence interval is:\[64.64 \pm 0.793\]Which gives the interval:\[(63.847, 65.433)\]

02

Calculate the 95% Confidence Interval for Left-Handers

Repeat the same procedure for the strong left-handers. Use the same confidence interval formula with \(\bar{x} = 63.97\), \(s = 15.4\), \(n = 236\), and \(z = 1.96\). Calculating the margin of error:\[\text{ME} = 1.96 \times \left(\frac{15.4}{\sqrt{236}}\right) \approx 1.963\]Thus, the confidence interval is:\[63.97 \pm 1.963\]Which gives the interval:\[(62.007, 65.933)\]

03

Compare Widths of the Confidence Intervals

The confidence interval for right-handers is narrower than for left-handers: (63.847, 65.433) vs. (62.007, 65.933). This is because the sample size for right-handers (1472) is much larger than for left-handers (236), reducing the standard error and thus the width of the confidence interval.

04

Determine if There is a Difference in Means

To determine if there is a significant difference, check if the confidence intervals for the two groups overlap. Since they overlap in the range \([63.847, 65.433]\) and \([62.007, 65.933]\), you cannot conclude that there is a significant difference in mean ages of death for left- and right-handers based solely on these results.

05

Compute Confidence Interval for Difference in Means

Calculate the confidence interval for the difference in means using the difference \(d = \bar{x}_R - \bar{x}_L = 64.64 - 63.97 = 0.67\). The standard error for the difference is:\[SE = \sqrt{\left(\frac{15.5^2}{1472}\right) + \left(\frac{15.4^2}{236}\right)} \approx 1.023\]The confidence interval for the difference is:\[0.67 \pm 1.96 \times 1.023\approx 0.67 \pm 2.005\]This gives the interval:\[(-1.335, 2.675)\]

06

Interpret the Result

The confidence interval for the difference between the means includes zero \((-1.335, 2.675)\), indicating there is no significant difference in the mean ages of death for strong right-handers and left-handers at the 95% confidence level.

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.

Confidence Interval

Confidence intervals play a crucial role in statistical analysis. They give us a range in which we expect the true population parameter, like the mean, to lie with a certain level of confidence, often expressed as a percentage. In our case, we want to know the age at which strong right-handed or left-handed baseball players typically pass away, with 95% confidence. To compute a 95% confidence interval, we use the formula: \[CI = \bar{x} \pm z \left(\frac{s}{\sqrt{n}}\right)\]where:

• \(\bar{x}\) = Sample mean
• \(z\) = Z-score (approximately 1.96 for 95% confidence)
• \(s\) = Sample standard deviation
• \(n\) = Sample size

For right-handers, the confidence interval tells us we can be 95% confident their mean age falls between 63.847 and 65.433 years. Similarly, for left-handers, it likely falls between 62.007 and 65.933 years. These intervals provide insight but do not give a specific age, adding to our understanding of variability among samples.

Sample Size

Sample size, denoted by \(n\), is a fundamental aspect when calculating confidence intervals. It directly influences the accuracy of the interval. A larger sample size results in a narrower confidence interval, which indicates a more precise estimate.In the exercise, the sample size for right-handed players is 1472, while for left-handed players it is only 236. This discrepancy leads to a more confident and narrower confidence interval for the former group due to their larger sample size. Key Importance of Sample Size:

• Larger sample sizes reduce the standard error. This provides a more accurate representation of the population parameter.
• Small sample sizes may lead to larger confidence intervals, reducing precision.

Therefore, when comparing the two groups, it's clear that the confidence interval for right-handers appears tighter due to their substantial sample size.

Standard Deviation

Standard deviation, represented as \(s\), measures the amount of variation or dispersion in a set of values. In statistical analysis, standard deviation helps us understand how spread out sample data points are relative to the mean.In this context, both right and left-handed players have similar standard deviations: 15.5 and 15.4, respectively. This similarity indicates that the variability in ages at which players pass away is quite comparable between the two groups. Why Does Standard Deviation Matter?

• It affects the "width" of the confidence interval.
• Lower standard deviations mean data points are closer to the mean, leading to tighter confidence intervals.
• Higher standard deviations suggest more variability, which can stretch the confidence interval.

Despite having different sample sizes, the similar standard deviations highlight that the inherent variability in the datasets is consistent.

Mean Comparison

Mean comparison involves assessing whether there is a statistically significant difference between the means of two groups. In this scenario, it examines if right- and left-handed baseball players differ in their average age of death.The results show overlapping confidence intervals for right-handers \(63.847, 65.433\) and left-handers \(62.007, 65.933\). Because of this overlap, we cannot confidently say there is a difference in their mean age of death; it suggests the means of these two groups could be close.Further Insight:

• The computed confidence interval for the difference in means is \(-1.335, 2.675\). Since this interval includes zero, it supports the conclusion that any observed difference might be due to chance.
• If intervals did not overlap, and the interval for their difference excluded zero, we might infer a significant difference in means.

This analysis shows the critical nature of confidence intervals and sample comparisons in determining true differences between groups.