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 7.1.37a

Chapter 7, Problem 7.1.37a

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.

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. Since the confidence level is 90%, the corresponding z-score (Z) is approximately 1.645. The margin of error (E) is given as 3%, or 0.03. Since nothing is known about the population proportion, assume p = 0.5, which maximizes the required sample size.

Step 3: Substitute the values into the formula. Replace Z with 1.645, p with 0.5, and E with 0.03 in the formula: n = (1.645^2 * 0.5 * (1 - 0.5)) / 0.03^2.

Step 4: Simplify the numerator of the formula. Calculate 1.645^2, then multiply it by 0.5 and (1 - 0.5).

Step 5: Simplify the denominator of the formula. Calculate 0.03^2, then divide the simplified numerator by the simplified denominator to find the required sample size, n.

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Key Concepts

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

Sample Size Determination

Sample size determination is a statistical method used to calculate the number of observations or replicates needed in a survey to achieve a desired level of confidence and precision. In this case, the marketing team needs to determine how many consumers to survey to estimate the percentage of potential buyers accurately, ensuring that the estimate is within three percentage points of the true population percentage.

Confidence Level

The confidence level represents the degree of certainty that the sample accurately reflects the population. A 90% confidence level means that if the survey were conducted multiple times, 90% of the time the results would fall within the specified margin of error. This concept is crucial for understanding how reliable the survey results will be in predicting consumer behavior.

Margin of Error

The margin of error indicates the range within which the true population parameter is expected to lie, based on the sample results. In this scenario, a margin of error of three percentage points means that the estimated percentage of consumers intending to purchase the new iPhone could vary by plus or minus three percentage points from the actual percentage. This concept is essential for assessing the precision of the survey results.

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Finding the Minimum Sample Size Needed for a Confidence Interval

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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).

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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.

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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.)

a. If n independent trials result in no successes, why can’t we find confidence interval limits by using the methods described in this section?

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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.

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Textbook Question

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.

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