Question

Exercise 8.106 presented statistics from a study of fast starts by ice hockey skaters. The mean and standard deviation of the 69 individual average acceleration measurements over the 6 -meter distance were 2.962 and .529 meters per second, respectively. a. Find a \(95 \%\) confidence interval for this population mean. Interpret the interval. b. Suppose you were dissatisfied with the width of this confidence interval and wanted to cut the interval in half by increasing the sample size. How many skaters (total) would have to be included in the study?

Short answer

Answer: The 95% confidence interval for the population mean of acceleration measurements is (2.838, 3.086) meters per second. To cut the width of the confidence interval in half, 250 skaters would have to be included in the study.

Step-by-step solution

Find the Z-score for the given level of confidence

Since we are asked to find a 95% confidence interval, the level of significance, \(\alpha\), is 1 - 0.95 = 0.05. We will now find the Z-score, \(Z_{\frac{\alpha}{2}}\), that corresponds to the point in the standard normal distribution such that 95% of the area is between both tails: \(Z_{0.025} = 1.96\)

Calculate the 95% confidence interval

We have all the necessary data to calculate the confidence interval using the formula: CI = \(\bar{x} \pm (Z_{\frac{\alpha}{2}} \times \frac{s}{\sqrt{n}})\) CI = \(2.962 \pm (1.96 \times \frac{0.529}{\sqrt{69}})\) Now, we calculate the confidence interval: CI = \(2.962 \pm 0.124\) Thus, the 95% confidence interval for the population mean is (2.838, 3.086).

Cut the interval in half and find the new sample size

To cut the width of the confidence interval in half, we will keep the same formula and proportions, but we will use a new sample size (let's call it n_new): \(\frac{1.96 \times \frac{0.529}{\sqrt{n_{new}}}} = \frac{1}{2} \times (0.248)\) Now, we need to solve for \(n_{new}\): \(n_{new} = (\frac{1.96 \times 0.529}{0.124})^2 = 249.47\) Since we can't have a fraction of a skater, we need to round up to the nearest whole number, so the new sample size required would be: \(n_{new} = 250\)

Conclusion

a. The \(95 \%\) confidence interval for the population mean of acceleration measurements is (2.838, 3.086) meters per second. b. To cut the width of the confidence interval in half, a total of 250 skaters would have to be included in the study.

Key concepts

Normal Distribution

The normal distribution, also known as the Gaussian distribution, is a probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean. In graph form, this distribution will appear as a bell-shaped curve. The bell curve has large implications in statistical analysis. It comes into play, especially when discussing samples and population parameters like the mean.

When a dataset is normally distributed, we can make certain assumptions about the probabilities of data points lying within certain ranges. For example, about 68% of values drawn from a normal distribution are within one standard deviation away from the mean; 95% are within two standard deviations; and 99.7% are within three standard deviations. This rule is called the Empirical Rule or 68-95-99.7 rule and is key when working with confidence intervals.

Standard Deviation

Standard deviation is a measure of the dispersion or variation in a set of values. It indicates how much individual measurements in a dataset typically deviate from the mean or average value of that set. For instance, a high standard deviation means that the data points are spread out over a wider range of values and are less consistent, while a low standard deviation indicates that the data points are generally close to the mean.

Understanding standard deviation is crucial when estimating the confidence interval because it directly affects the width of the interval. If you take the standard deviation into account along with the size of your sample, you can anticipate the range in which your population mean likely lies. The smaller the standard deviation, the narrower (and thus more precise) your confidence interval will be.

Sample Size Calculation

The calculation of sample size is a fundamental aspect of designing an experiment and conducting statistical inference. The sample size affects the precision of your estimates; generally, the larger the sample size, the more precise your estimates will be. Statisticians use sample size calculations to determine the number of observations or replicates required to detect an effect of a given size with a desired degree of confidence and power.

Calculating the appropriate sample size involves understanding the desired confidence level, the margin of error (width of the confidence interval), the expected variability in the data (standard deviation), and the population size. Larger sample sizes result in more reliable and precise estimations of population parameters but also increase the cost and effort to study the population. Therefore, it's a balance between practicality and statistical aims.

Population Mean

The population mean is the average of a set of characteristics (e.g., measurements, scores, etc.) within a population. It is a parameter that is estimated using sample data. As in many cases, the entire population can't be measured directly; statisticians take a smaller, representative sample from which they can make inferences about the population mean.

It's important to note that while the mean of the sample is an unbiased estimate of the population mean, there are always errors associated with the estimation process. This is why confidence intervals are used—to express the degree of uncertainty surrounding this population mean estimate. The width of the confidence interval is affected by the variability of the data and the size of the sample, but it also depends on the confidence level you seek. Higher confidence levels lead to wider intervals, implying less certainty about the precision of the population mean estimate.