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 7.1.32b

Chapter 7, Problem 7.1.32b

Touch Your Nose With Your Tongue Find the sample size needed to estimate the percentage of adults who can touch their nose with their tongue. Use a margin of error of 2 percentage points and use a confidence level of 90%.

b. Assume that a previous study showed that 10% of adults can touch their nose with their tongue (based on data from Onedio).

Verified step by step guidance

Step 1: Identify the formula for determining the sample size for estimating a population proportion. The formula is:

n = (z² × p × (1 - p)) / E²

, where

n

is the sample size,

z

is the z-score corresponding to the confidence level,

p

is the estimated proportion, and

E

is the margin of error.

Step 2: Determine the z-score for a 90% confidence level. The z-score corresponds to the critical value from the standard normal distribution. For a 90% confidence level, the z-score is approximately 1.645.

Step 3: Substitute the given values into the formula. The estimated proportion

p

is 0.10 (10%), the margin of error

E

is 0.02 (2 percentage points), and the z-score is 1.645.

Step 4: Perform the calculation step-by-step. First, calculate

p × (1 - p)

, which is

0.10 × (1 - 0.10) = 0.10 × 0.90 = 0.09

. Then, calculate

z²

, which is

1.645² = 2.706025

. Finally, divide the product of

z² × p × (1 - p)

E²

, where

E² = 0.02² = 0.0004

Step 5: Interpret the result. The calculated value of

n

will give the minimum sample size required to estimate the percentage of adults who can touch their nose with their tongue within the specified margin of error and confidence level.

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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 the results of a study are reliable and valid. It takes into account the desired margin of error, confidence level, and the estimated proportion of the population. In this case, the sample size will help estimate the percentage of adults who can touch their nose with their tongue with a specified accuracy.

Margin of Error

The margin of error is a statistic that expresses the amount of random sampling error in a survey's results. It indicates the range within which the true population parameter is expected to fall, given a certain confidence level. For example, a margin of error of 2 percentage points means that if the survey result is 10%, the true percentage is likely between 8% and 12%.

Confidence Level

The confidence level is the probability that the value of a parameter falls within a specified range of values. Common confidence levels are 90%, 95%, and 99%. A 90% confidence level means that if the same population were sampled multiple times, approximately 90% of the calculated confidence intervals would contain the true population parameter, providing a measure of reliability for the estimate.

Recommended video:

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Introduction to Confidence Intervals

Related Practice

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.

Touch Therapy When she was 9 years of age, Emily Rosa did a science fair experiment in which she tested professional touch therapists to see if they could sense her energy field. She flipped a coin to select either her right hand or her left hand, and then she asked the therapists to identify the selected hand by placing their hand just under Emily’s hand without seeing it and without touching it. Among 280 trials, the touch therapists were correct 123 times (based on data in “A Close Look at Therapeutic Touch,” Journal of the American Medical Association, Vol. 279, No. 13).

c. Using Emily’s sample results, construct a 99% confidence interval estimate of the proportion of correct responses made by touch therapists.

Textbook Question

Mean Pulse Rate of Males Data Set 1 “Body Data” in Appendix B includes pulse rates of 153 randomly selected adult males, and those pulse rates vary from a low of 40 bpm to a high of 104 bpm. Find the minimum sample size required to estimate the mean pulse rate of adult males. Assume that we want 99% confidence that the sample mean is within 2 bpm of the population mean.

b. Assume that sigma=11.3 bpm, based on the value of s=11.3 bpm for the sample of 153 male pulse rates.

Textbook Question

Astrology A sociologist plans to conduct a survey to estimate the percentage of adults who believe in astrology. How many people must be surveyed if we want a confidence level of 99% and a margin of error of four percentage points?

b. Use the information from a previous Harris survey in which 26% of respondents said that they believed in astrology.

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.

Tennis Challenges In a recent U. S. Open tennis tournament, women playing singles matches used challenges on 137 calls made by the line judges. Among those challenges, 33 were found to be successful with the call overturned.

b. Compare the result from part (a) to this 99% confidence interval for the percentage of successful challenges made by the men playing singles matches: . Does it appear that either gender is more successful than the other?

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.

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c. Repeat part (b) using a confidence level of 80%.

Textbook Question

Women Who Give Birth An epidemiologist plans to conduct a survey to estimate the percentage of women who give birth. How many women must be surveyed in order to be 99% confident that the estimated percentage is in error by no more than two percentage points?

c. What is wrong with surveying randomly selected adult women?