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 7.2.26b

Chapter 7, Problem 7.2.26b

Mean Pulse Rate of Females Data Set 1 “Body Data” in Appendix B includes pulse rates of 147 randomly selected adult females, and those pulse rates vary from a low of 36 bpm to a high of 104 bpm. Find the minimum sample size required to estimate the mean pulse rate of adult females. Assume that we want 99% confidence that the sample mean is within 2 bpm of the population mean.

b. Assume that sigma=12.5 bpm, based on the value of s=12.5 bpm for the sample of 147 female pulse rates.

Verified step by step guidance

Step 1: Identify the formula for determining the minimum sample size required to estimate the population mean. The formula is:

n = \frac{{(z)}^{2}}{σ^{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 values given in the problem. The confidence level is 99%, so the z-score corresponding to this confidence level is approximately 2.576. The population standard deviation

σ

is given as 12.5 bpm, and the margin of error

E

is 2 bpm.

Step 3: Substitute the known values into the formula. Replace

z

with 2.576,

σ

with 12.5, and

E

with 2 in the formula:

n = \frac{{(2.576)}^{2}}{{12.5}^{2}}

Step 4: Simplify the numerator by squaring the z-score and the standard deviation. Then, simplify the denominator by squaring the margin of error.

Step 5: Divide the simplified numerator by the simplified denominator to calculate the minimum sample size. If the result is not a whole number, always 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 Determination

Sample size determination is a statistical process used to calculate the number of observations needed to achieve a desired level of precision in estimating a population parameter. In this context, it involves using the desired confidence level and margin of error to ensure that the sample mean is a reliable estimate of the population mean.

Confidence Interval

A confidence interval is a range of values, derived from a sample, that is likely to contain the population parameter with a specified level of confidence. In this case, a 99% confidence level indicates that if we were to take many samples, approximately 99% of the calculated intervals would contain the true population mean.

Standard Deviation and Population Variance

Standard deviation is a measure of the amount of variation or dispersion in a set of values. In this scenario, the given standard deviation (sigma = 12.5 bpm) is crucial for calculating the sample size, as it reflects the variability in pulse rates among adult females, influencing the precision of the estimate.

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Population Standard Deviation Unknown

Related Practice

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).

b. How does the result compare to the confidence interval found in Exercise 14 in Section 7-3?

Textbook Question

Online Gambling Some states now allow online gambling. As a marketing manager for a casino, you need to determine the percentage of adults in those states who gamble online. How many adults must you survey in order to be 99% confident that your estimate is in error by no more than two percentage points?

b. Assume that 18% of all adults gamble online (based on 2017 data from a Gambling Commission study in Great Britain).

Textbook Question

Minting Quarters Listed below are weights (grams) of quarters minted after 1964 (based on Data Set 40 “Coin Weights” in Appendix B).

b. Specifications require that the quarters have a weight of 5.670 g. What does the confidence interval suggest about that specification?

Textbook Question

7. FRESHMAN 15 Here is a sample of amounts of weight change (kg) of college students in their freshman year (from Data Set 13 “Freshman 15” in Appendix B): 11, 3, 0, –2, where –2 represents a loss of 2 kg and positive values represent weight gained. Here are ten bootstrap samples:

{11, 11, 11, 0}, {11, –2, 0, 11}, {11, –2, 3, 0}, {3, –2, 0, 11}, {0, 0, 0, 3}, {3, –2, 3, –2}, {11, 3, –2, 0}, {–2, 3, –2, 3}, {–2, 0, –2, 3}, {3, 11, 11, 11}.

b. Using only the ten given bootstrap samples, construct an 80% confidence interval estimate of the standard deviation of the weight changes for the population.

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?

b. Assume that 11% of consumers have a smartphone and plan to upgrade to a new model.

Textbook Question

Caffeine in Soft Drinks Listed below are measured amounts of caffeine (mg per 12 oz of drink) obtained in one can from each of 20 brands (7UP, A&W Root Beer, Cherry Coke, . . . , TaB).

b. Given that Exercise 20 in Section 7-2 used the same data for a 99% confidence interval based on use of the t distribution, and given that the data do not appear to be from a normally distributed population, which confidence interval is likely to be better: The confidence interval from part (a) or the confidence interval found in Exercise 20 in Section 7-2?