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 7.1.33b

Chapter 7, Problem 7.1.33b

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.

Verified step by step guidance

Step 1: Identify the formula for determining the required sample size for estimating a population proportion. The formula is: n = (Z^2 * p * (1 - p)) / E^2, where n is the required sample size, Z is the z-score corresponding to the desired confidence level, p is the estimated population proportion, and E is the margin of error.

Step 2: Determine the values for the variables in the formula. From the problem, the confidence level is 95%, so the z-score (Z) is approximately 1.96. The estimated population proportion (p) is 3.7% or 0.037, and the margin of error (E) is 1.5 percentage points or 0.015.

Step 3: Substitute the values into the formula. Replace Z with 1.96, p with 0.037, and E with 0.015 in the formula: n = (1.96^2 * 0.037 * (1 - 0.037)) / 0.015^2.

Step 4: Simplify the numerator of the formula. Calculate Z^2 (1.96^2), p * (1 - p) (0.037 * (1 - 0.037)), and multiply these values together.

Step 5: Simplify the denominator of the formula. Calculate E^2 (0.015^2), and then divide the numerator by the denominator to find the required sample size (n). Round up to the nearest whole number, as sample size must be an integer.

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Key Concepts

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

Sample Size Calculation

Sample size calculation is a statistical method used to determine the number of observations or replicates needed to ensure that a survey or experiment can yield reliable results. It takes into account the desired confidence level, margin of error, and the estimated proportion of the population exhibiting a certain characteristic. In this case, the calculation will help determine how many adults need to be surveyed to accurately estimate the current e-cigarette usage rate.

Confidence Level

The confidence level represents the degree of certainty that the true population parameter lies within the confidence interval calculated from the sample data. A 95% confidence level means that if the same survey were conducted multiple times, approximately 95% of the calculated intervals would contain the true proportion of e-cigarette users. This concept is crucial for understanding the reliability of the survey results.

Margin of Error

Margin of error quantifies the uncertainty associated with survey results, indicating the range within which the true population parameter is expected to fall. A margin of error of 1.5 percentage points means that the estimated proportion of e-cigarette users could be 1.5% higher or lower than the reported figure. This concept is essential for assessing the precision of the survey findings and guiding the sample size determination.

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Finding the Minimum Sample Size Needed for a Confidence Interval

Related Practice

Textbook Question

Wiggle Your Ears Find the sample size needed to estimate the percentage of adults who can wiggle their ears. Use a margin of error of 3 percentage points and use a confidence level of 99%.

b. Assume that 22% of adults can wiggle their ears (based on data from Soul Publishing).

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

Finite Population Correction Factor If a simple random sample of size n is selected without replacement from a finite population of size (n>0.05N), and the sample size is more than 5% of the population size , better results can be obtained by using the finite population correction factor, which involves multiplying the margin of error E by [Image]. Refer to the weights of the M&M candies in Data Set 38 “Candies” in Appendix B.

b. Use only the red M&Ms and treat that sample as a simple random sample selected from the population of the 345 M&Ms listed in the data set. Find the 95% confidence interval estimate of the mean weight of all 345 M&Ms. Compare the result to the actual mean of the population of all 345 M&Ms.

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

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?

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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.