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Ch. 7 - Estimating Parameters and Determining Sample Sizes
Triola - Elementary Statistics 14th Edition
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Tutti i libri di testoTriola 14th EditionCh. 7 - Estimating Parameters and Determining Sample SizesProblema 7.2.33
Capitolo 7, Problema 7.2.33

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.


"Table showing age distribution of federal prisoners: 16-25 (13), 26-35 (61), 36-45 (66), 46-55 (36), 56-65 (14), Over 65 (5)."

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Step 1: Calculate the midpoint for each age group. The midpoint is the average of the lower and upper bounds of each age group. For example, for the age group 16–25, the midpoint is (16 + 25) / 2 = 20.5.
Step 2: Multiply the midpoint of each age group by the corresponding frequency (number of inmates) to find the weighted contribution of each group to the total sum. For example, for the age group 16–25, the contribution is 20.5 × 13.
Step 3: Sum up all the weighted contributions from Step 2 to calculate the total sum of ages. Also, sum up all the frequencies to find the total number of inmates.
Step 4: Divide the total sum of ages by the total number of inmates to calculate the sample mean age. This is the point estimate for the mean age.
Step 5: Use the formula for the standard error of the mean (SE = s / √n, where s is the sample standard deviation and n is the sample size) and the t-distribution to construct the 95% confidence interval. The confidence interval is given by: Mean ± (t * SE), where t is the critical value from the t-distribution table for 95% confidence and degrees of freedom (df = n - 1).

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Confidence Interval

A confidence interval is a range of values, derived from sample statistics, that is likely to contain the true population parameter. For example, a 95% confidence interval suggests that if we were to take many samples and construct intervals in the same way, approximately 95% of those intervals would contain the true mean. This concept is crucial for estimating population parameters based on sample data.
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Introduction to Confidence Intervals

Mean

The mean, or average, is a measure of central tendency that is calculated by summing all values in a dataset and dividing by the number of values. In the context of the ages of prisoners, the mean age provides a single value that represents the central point of the age distribution, which is essential for constructing the confidence interval.
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Calculating the Mean

Frequency Distribution

A frequency distribution is a summary of how often each value occurs in a dataset. In this case, the table shows the number of prisoners within specific age ranges. Understanding frequency distributions is important for analyzing data, as it helps identify patterns and informs the calculation of statistics like the mean and confidence intervals.
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Intro to Frequency Distributions
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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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Confidence Levels

Given specific sample data, such as the data given in Exercise 1, which confidence interval is wider: the 95% confidence interval or the 80% confidence interval? Why is it wider?

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Los Angeles Commute Time Listed below are 15 Los Angeles commute times (based on a sample from Data Set 31 “Commute Times” in Appendix B). Construct a 99% confidence interval estimate of the population mean. Is the confidence interval a good estimate of the population mean?


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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?

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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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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?

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