Ch. 7 - Estimating Parameters and Determining Sample Sizes

Triola - Elementary Statistics 14th Edition

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Triola 14th Edition

Ch. 7 - Estimating Parameters and Determining Sample Sizes

Problem 7.2.13

Chapter 7, Problem 7.2.13

Archeology Archeologists have studied sizes of Egyptian skulls in an attempt to determine whether breeding occurred between different cultures. Listed below are the widths (mm) of skulls from 150 A.D. (based on data from Ancient Races of the Thebaid by Thomson and Randall-Maciver). Construct a 99% confidence interval estimate of the mean skull width.

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Step 1: Calculate the sample mean (\( \bar{x} \)) of the given skull widths. Add all the skull widths together and divide by the total number of data points. The formula is \( \bar{x} = \frac{\sum x_i}{n} \), where \( x_i \) are the individual data points and \( n \) is the sample size.

Step 2: Calculate the sample standard deviation (\( s \)) using the formula \( s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n-1}} \). Subtract the mean from each data point, square the result, sum these squared differences, divide by \( n-1 \), and then take the square root.

Step 3: Determine the critical value (\( t^* \)) for a 99% confidence level using a t-distribution table. The degrees of freedom (df) are \( n-1 \). For this problem, \( n = 8 \), so \( df = 7 \). Look up the critical value corresponding to a 99% confidence level and \( df = 7 \).

Step 4: Calculate the margin of error (ME) using the formula \( ME = t^* \cdot \frac{s}{\sqrt{n}} \). Multiply the critical value by the standard error, which is the sample standard deviation divided by the square root of the sample size.

Step 5: Construct the confidence interval using the formula \( \text{Confidence Interval} = \bar{x} \pm ME \). Add and subtract the margin of error from the sample mean to find the lower and upper bounds of the confidence interval.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Confidence Interval

A confidence interval is a range of values, derived from sample statistics, that is likely to contain the population parameter with a specified level of confidence. For example, a 99% confidence interval suggests that if we were to take many samples and construct intervals in the same way, approximately 99% of those intervals would contain the true population mean.

Mean

The mean, or average, is a measure of central tendency calculated by summing all the values in a dataset and dividing by the number of values. It provides a single value that represents the center of the data distribution, making it easier to understand the overall trend of the data, such as the average skull width in this case.

Standard Deviation

Standard deviation is a statistic that measures the dispersion or spread of a set of values around the mean. A low standard deviation indicates that the values tend to be close to the mean, while a high standard deviation indicates a wider spread. It is crucial for calculating the confidence interval, as it helps determine how much variability exists in the skull width measurements.

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Textbook Question

Cell Phone Radiation. Listed below are amounts of cell phone radiation (W/kg) measured from randomly selected cell phones (based on data from the Federal Communications Commission). Use these values for Exercises 1–6.

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Level of Measurement What is the level of measurement of these data (nominal, ordinal, interval, ratio)? Are the original unrounded amounts of radiation continuous data or discrete data?

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Textbook Question

Ages of Prisoners The accompanying frequency distribution summarizes sample data consisting of ages of randomly selected inmates in federal prisons (based on data from the Federal Bureau of Prisons). Use the data to construct a 95% confidence interval estimate of the mean age of all inmates in federal prisons.

Textbook Question

Estimating the Median Use the sample data listed in Exercise 1 “Bootstrap Requirements” to generate 1000 bootstrap samples, and find the median in each of those samples. After obtaining the 1000 sample medians, find the 95% confidence interval estimate of the population median by evaluating p2.5 and p97.5 from the sorted 1000 medians. Given that the sample times in Exercise 1 are from the 50 times in Data Set 20 “Alcohol and Tobacco in Movies” and those 50 times have a median of 5.5, how well did the bootstrap method work to create a “good” confidence interval?

Textbook Question

Requirements A construction quality control analyst has collected a random sample of six concrete road barriers, and she plans to weigh each of them and construct a 95% confidence interval estimate of the mean weight of all such barriers. What requirements must be satisfied in order to construct the confidence interval with the method from Section 7-2 that uses the t distribution?

Textbook Question

Degrees of Freedom In general, what does “degrees of freedom” refer to? For the sample data described in Exercise 7 “Requirements,” find the number of degrees of freedom, assuming that you want to construct a confidence interval estimate of u using the t distribution.

Textbook Question

Mean Body Temperature Data Set 5 “Body Temperatures” in Appendix B includes a sample of 106 body temperatures having a mean of 98.20 F and a standard deviation of 0.62 F. Construct a 95% confidence interval estimate of the mean body temperature for the entire population. What does the result suggest about the common belief that 98.6 F is the mean body temperature?