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 7.1.42b

Chapter 7, Problem 7.1.42b

No Failures According to the Rule of Three, when we have a sample size n with x=0 successes, we have 95% confidence that the true population proportion has an upper bound of 3/n. (See “A Look at the Rule of Three,” by Jovanovic and Levy, American Statistician, Vol. 51, No. 2.)

b. In a study of failure rates of computer hard drives, 45 Toshiba model MD04ABA500V hard drives were tested and there were no failures. What is the 95% upper bound for the percentage of failures for the population of all such hard drives?

Verified step by step guidance

Step 1: Understand the Rule of Three. The Rule of Three states that when we have a sample size n and observe x=0 successes (or failures in this case), the 95% confidence upper bound for the true population proportion is given by the formula:

\frac{3}{n}

Step 2: Identify the given values from the problem. Here, the sample size n is 45, and the number of observed failures x is 0.

Step 3: Substitute the value of n into the formula for the upper bound. The formula becomes:

\frac{3}{45}

Step 4: Simplify the fraction to find the upper bound for the proportion of failures. This will give the proportion in decimal form.

Step 5: Convert the proportion to a percentage by multiplying the result by 100. This will provide the 95% upper bound for the percentage of failures in the population of all such hard drives.

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Key Concepts

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

Rule of Three

The Rule of Three is a statistical principle used when no successes are observed in a sample. It states that if a sample of size n has x=0 successes, we can be 95% confident that the true proportion of successes in the population is less than or equal to 3/n. This rule is particularly useful in estimating upper bounds for proportions in small sample sizes.

Confidence Interval

A confidence interval is a range of values, derived from sample statistics, that is likely to contain the true population parameter. The 95% confidence level indicates that if we were to take many samples and build intervals, approximately 95% of those intervals would contain the true parameter. In the context of the Rule of Three, it helps quantify uncertainty about the population proportion based on observed data.

Sample Size

Sample size refers to the number of observations or data points collected in a study. It plays a crucial role in statistical analysis, as larger sample sizes generally provide more reliable estimates of population parameters. In the Rule of Three, the sample size (n) directly influences the upper bound of the failure rate, with smaller samples leading to wider confidence intervals.

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Textbook Question

Smart Phone Apple is planning for the launch of a new and improved iPhone. The marketing team wants to know the worldwide percentage of consumers who intend to purchase the new model, so a survey is being planned. How many people must be surveyed in order to be 90% confident that the estimated percentage is within three percentage points of the true population percentage?

a. Assume that nothing is known about the worldwide percentage of consumers who intend to buy the new model.

Textbook Question

Large Data Sets from Appendix B. In Exercises 21 and 22, use the data set in Appendix B. Assume that each sample is a simple random sample obtained from a population with a normal distribution.

Birth Weights Refer to Data Set 6 “Births” in Appendix B.

a. Use the 205 birth weights of girls to construct a 95% confidence interval estimate of the standard deviation of the population from which the sample was obtained.

Textbook Question

15. HEIGHTS OF FEMALE SOCCER PLAYERS Listed below are the heights (in.) of players on the U.S. Women’s National Soccer Team (at the time of this writing). Use those heights as a sample of the heights of all professional women soccer players.

a. Use 1000 bootstrap samples to construct a 95% confidence interval estimate of σ.

Textbook Question

E-Cigarettes A New York Times article reported that a survey conducted in 2014 included 36,000 adults, with 3.7% of them being regular users of e-cigarettes. Because e-cigarette use is relatively new, there is a need to obtain today’s usage rate. How many adults must be surveyed now if we want a confidence level of 95% and a margin of error of 1.5 percentage points?

b. Use the results from the 2014 survey.

Textbook Question

Voting Survey In a survey of 1002 people, 70% said that they voted in a recent presidential election (based on data from ICR Research Group). Voting records show that 61% of eligible voters actually did vote.

b. Find a 95% confidence interval estimate of the percentage of people who say that they voted.

Textbook Question

Mean Body Temperature Data Set 5 “Body Temperatures” in Appendix B includes 106 body temperatures of adults for Day 2 at 12 AM, and they vary from a low of 96.5F to a high of 99.6F. Find the minimum sample size required to estimate the mean body temperature of all adults. Assume that we want 98% confidence that the sample mean is within 0.1F of the population mean.

b. Assume that sigma=0.62F, based on the value of s=0.62F for the sample of 106 body temperatures.