Ch. 7 - Estimating Parameters and Determining Sample Sizes

Triola - Elementary Statistics 14th Edition

Triola14th EditionElementary StatisticsISBN: 9780137366446Not the one you use?Change textbook

Triola 14th Edition

Ch. 7 - Estimating Parameters and Determining Sample Sizes

Problem 7.6

Chapter 7, Problem 7.6

Sample Size for Mean Find the sample size required to estimate the mean IQ of airline pilots. Assume that we want 99% confidence that the mean from the sample is within two IQ points of the true population mean. Also assume that sigma=15

Verified step by step guidance

Step 1: Identify the formula for sample size estimation when estimating a population mean. The formula is:

n = (z * σ / E)

², where n is the sample size, z is the z-score corresponding to the confidence level, σ is the population standard deviation, and E is the margin of error.

Step 2: Determine the z-score for a 99% confidence level. For a 99% confidence level, the z-score corresponds to the critical value from the standard normal distribution. This value is approximately 2.576.

Step 3: Substitute the given values into the formula. Here, σ = 15 (population standard deviation) and E = 2 (margin of error). The formula becomes:

n = (2.576 * 15 / 2)

².

Step 4: Simplify the expression inside the parentheses. First, calculate the product of the z-score and the standard deviation, then divide by the margin of error.

Step 5: Square the result obtained in Step 4 to find the sample size. Round up to the nearest whole number if necessary, 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 in a study to achieve a desired level of precision. It involves considering factors such as the population standard deviation, the desired confidence level, and the margin of error. In this case, we need to calculate how many airline pilots' IQs should be sampled to ensure that our estimate is accurate within two points of the true mean.

Confidence Level

The confidence level represents the degree of certainty that the population parameter lies within a specified range of the sample estimate. A 99% confidence level indicates that if we were to take many samples, approximately 99% of the calculated confidence intervals would contain the true population mean. This high level of confidence requires a larger sample size to ensure that the estimate is reliable.

Margin of Error

The margin of error is the range within which the true population parameter is expected to fall, given a certain level of confidence. In this scenario, a margin of error of two IQ points means that we want our sample mean to be within two points of the actual mean IQ of airline pilots. This concept is crucial for determining how precise our estimate needs to be and directly influences the required sample size.

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Finding the Minimum Sample Size Needed for a Confidence Interval

Related Practice

Textbook Question

Female Motorcycle Owners Here is a 95% confidence interval estimate of the percentage of motorcycle owners who are female: 17.5%<p<20.6% (based on data from the Motorcycle Industry Council). What is the best point estimate of the percentage of motorcycle owners who are women?

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Textbook Question

Critical Thinking. In Exercises 17–28, use the data and confidence level to construct a confidence interval estimate of p, then address the given question.

Gender Selection Before its clinical trials were discontinued, the Genetics & IVF Institute conducted a clinical trial of the XSORT method designed to increase the probability of conceiving a girl and, among the 945 babies born to parents using the XSORT method, there were 879 girls. The YSORT method was designed to increase the probability of conceiving a boy and, among the 291 babies born to parents using the YSORT method, there were 239 boys. Construct the two 95% confidence interval estimates of the percentages of success. Compare the results. What do you conclude?

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Textbook Question

Red Blood Cell Count Here is a 95% confidence interval estimate of obtained by using the red blood cell counts of adult females listed in Data Set 1 “Body Data” in Appendix B:

[Image].

Identify the corresponding confidence interval estimate of and include the appropriate units.

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Textbook Question

FINDING SAMPLE SIZE Instead of using Table 7-2 for determining the sample size required to estimate a population standard deviation σ, the following formula can also be used

$n = \frac{1}{2} {(\frac{z_{α / 2}}{d})}^{2}$

where $z sub sub alpha divided by 2$ corresponds to the confidence level and d is the decimal form of the percentage error. For example, to be 95% confident that s is within 15% of the value of σ, use zα/2=1.96 and d=0.15 to get a sample size of n=86. Find the sample size required to estimate the standard deviation of IQ scores of data scientists, assuming that we want 98% confidence that s is within 5% of σ.

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Textbook Question

Determining Sample Size. Assume that each sample is a simple random sample obtained from a normally distributed population.

You want to estimate for the population of diastolic blood pressures of air traffic controllers in the United States. Find the minimum sample size needed to be 95% confident that the sample standard deviation s is within 1% of σ. Is this sample size practical?

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Textbook Question

Mint Specs Listed below are weights (grams) from a simple random sample of pennies produced after 1983 (from Data Set 40 “Coin Weights” in Appendix B). Construct a 95% confidence interval estimate of for the population of such pennies. What does the confidence interval suggest about the U.S. Mint specifications that now require a standard deviation of 0.0230 g for weights of pennies?

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