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Chapter 9: Problem 43

A study of the ability of individuals to walk in a straight line ("Can We Really Walk Straight?" American Journal of Physical Anthropology [1992]: \(19-27\) ) reported the following data on cadence (strides per second) for a sample of \(n=20\) randomly selected healthy men: \(\begin{array}{llllllllll}0.95 & 0.85 & 0.92 & 0.95 & 0.93 & 0.86 & 1.00 & 0.92 & 0.85 & 0.81\end{array}\) \(\begin{array}{llllllllll}0.78 & 0.93 & 0.93 & 1.05 & 0.93 & 1.06 & 1.06 & 0.96 & 0.81 & 0.96\end{array}\) Construct and interpret a \(99 \%\) confidence interval for the population mean cadence.

Short Answer

Expert verified
The 99% confidence interval for the population mean cadence would be the result from step 4. These are the values within which we can be 99% confident that the true population mean lies.

Step by step solution

01

Calculation of Mean

Firstly, the mean (average) of these strides per seconds values needs to be calculated. The mean \(\mu\) is calculated as the sum of all values divided by the number of values.
02

Calculation of Standard Deviation

Next, the standard deviation is computed. The standard deviation \(s\) is found by calculating the square root of the variance. Variance is the average of the squared differences from the mean.
03

Determining the t-distribution value

Then, search for the t-value that corresponds to a 99% confidence level and 19 degrees of freedom (n-1) from the t-distribution table.
04

Calculation of Confidence Interval

Finally, the confidence interval is calculated. The formula for the confidence interval is \(\mu \pm (t * \frac{s}{\sqrt{n}})\), where \(\mu\) is mean, \(t\) is the t-value from step 3, \(s\) is the standard deviation, and \(n\) is the number of observations. This is going to give the lower and upper limits of the confidence interval.

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Key Concepts

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

Mean Calculation
The mean is a fundamental concept in statistics, representing the average value of a dataset. To calculate the mean, sum all the values and divide by the total number of values.
For the given data, this involves adding up all cadence values and dividing by 20, the total number of subjects.
  • Mean (\(ar{x}\)) = \(\frac{\text{Sum of all cadence values}}{20}\)
In this exercise, once you've calculated the sum, you divide by 20 to find the mean cadence. This mean provides a baseline understanding of the general cadence rate among the studied men.
Standard Deviation
Standard deviation is a measure that describes how much individual measurements deviate from the mean. It's crucial for understanding the spread of data.
First, calculate the variance, which is the average of the squared differences from the mean, and then take the square root to find the standard deviation.
  • Step 1: Find the difference of each data point from the mean.
  • Step 2: Square each difference.
  • Step 3: Compute the average of these squared differences (this is the variance).
  • Step 4: Take the square root of the variance to get the standard deviation \(s\).
Getting this measure helps us to tell how much variation exists in the cadence data, providing a deeper understanding of the consistency among individuals.
t-distribution
The t-distribution is a statistical distribution that is particularly useful when dealing with small sample sizes (less than 30). It's similar to the normal distribution but accounts for more variability, especially with limited data.
For our scenario, with a sample size of 20, we use the t-distribution to find our critical t-value.
  • This requires looking up the t-value in a table for a 99% confidence level with \(n-1\) degrees of freedom (here, 19).
The resulting t-value allows us to determine the range and reliability of our confidence interval, adapting for the smaller sample size.
Inference
Inference in statistics is about drawing conclusions from data. Once you have the confidence interval, you use it for inferential statistics to make predictions or decisions about a population.
For the 99% confidence interval calculated for the mean cadence, it implies that if we were to take multiple samples and compute a confidence interval for each, 99% of those intervals would contain the true population mean.
  • This helps to express how confident one is about the estimate of the population parameter.
It allows you to make predictions and decisions with an understood level of uncertainty, foundational for statistical analysis and research.

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Most popular questions from this chapter

The article "Consumers Show Increased Liking for Diesel Autos" (USA Today, January 29,2003 ) reported that \(27 \%\) of U.S. consumers would opt for a diesel car if it ran as cleanly and performed as well as a car with a gas engine. Suppose that you suspect that the proportion might be different in your area and that you want to conduct a survey to estimate this proportion for the adult residents of your city. What is the required sample size if you want to estimate this proportion to within \(.05\) with \(95 \%\) confidence? Compute the required sample size first using 27 as a preliminary estimate of \(\pi\) and then using the conservative value of \(.5 .\) How do the two sample sizes compare? What sample size would you recommend for this study?

The interval from \(-2.33\) to \(1.75\) captures an area of \(.95\) under the \(z\) curve. This implies that another largesample \(95 \%\) confidence interval for \(\mu\) has lower limit \(\bar{x}-2.33 \frac{\sigma}{\sqrt{n}}\) and upper limit \(\bar{x}+1.75 \frac{\sigma}{\sqrt{n}} .\) Would you recommend using this \(95 \%\) interval over the \(95 \%\) interval \(\bar{x} \pm 1.96 \frac{\sigma}{\sqrt{n}}\) discussed in the text? Explain. (Hint: Look at the width of each interval.)

Samples of two different types of automobiles were selected, and the actual speed for each car was determined when the speedometer registered \(50 \mathrm{mph}\). The resulting \(95 \%\) confidence intervals for true average actual speed were \((51.3,52.7)\) and \((49.4,50.6)\). Assuming that the two sample standard deviations are identical, which confidence interval is based on the larger sample size? Explain your reasoning.

Why is an unbiased statistic generally preferred over a biased statistic for estimating a population characteristic? Does unbiasedness alone guarantee that the estimate will be close to the true value? Explain. Under what circumstances might you choose a biased statistic over an unbiased statistic if two statistics are available for estimating a population characteristic?

The Chronicle of Higher Education (January 13, 1993) reported that \(72.1 \%\) of those responding to a national survey of college freshmen were attending the college of their first choice. Suppose that \(n=500\) students responded to the survey (the actual sample size was much larger). a. Using the sample size \(n=500\), calculate a \(99 \%\) confidence interval for the proportion of college students who are attending their first choice of college. b. Compute and interpret a \(95 \%\) confidence interval for the proportion of students who are not attending their first choice of college. c. The actual sample size for this survey was much larger than 500 . Would a confidence interval based on the actual sample size have been narrower or wider than the one computed in Part (a)?
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