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 7.2.22

Chapter 7, Problem 7.2.22

Mean IQ of Data Scientists See the preceding exercise, in which we can assume that sigma=15 for the IQ scores. Data scientists are a group with IQ scores that vary less than the IQ scores of the general population. Find the sample size needed to estimate the mean IQ of data scientists, given that we want 98% confidence that the sample mean is within 2 IQ points of the population mean. Does the sample size appear to be practical?

Verified step by step guidance

Identify the formula for determining the required sample size for estimating a population mean: \( n = \left( \frac{z \cdot \sigma}{E} \right)^2 \), where \( n \) is the sample size, \( z \) is the z-score corresponding to the desired confidence level, \( \sigma \) is the population standard deviation, and \( E \) is the margin of error.

Determine the values for the variables in the formula: \( \sigma = 15 \) (given), \( E = 2 \) (the desired margin of error), and the confidence level is 98%, which corresponds to a z-score of approximately 2.33 (from a z-table or standard normal distribution).

Substitute the known values into the formula: \( n = \left( \frac{2.33 \cdot 15}{2} \right)^2 \).

Simplify the expression inside the parentheses: \( \frac{2.33 \cdot 15}{2} \), then square the result to calculate \( n \).

Evaluate whether the resulting sample size is practical by considering factors such as the feasibility of collecting data from the calculated number of participants and the resources available for the study.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

Video duration:

Was this helpful?

Key Concepts

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

Confidence Interval

A confidence interval is a range of values, derived from sample statistics, that is likely to contain the true population parameter. In this context, a 98% confidence level indicates that if we were to take many samples, approximately 98% of the calculated intervals would contain the true mean IQ of data scientists. This concept is crucial for understanding how sample size affects the precision of our estimates.

Sample Size Calculation

Sample size calculation involves determining the number of observations needed to achieve a desired level of precision in estimating a population parameter. The formula typically incorporates the desired confidence level, the population standard deviation (sigma), and the margin of error. In this case, we need to calculate how many data scientists' IQ scores are required to ensure that our estimate is within 2 IQ points of the true mean with 98% confidence.

Standard Deviation

Standard deviation is a measure of the amount of variation or dispersion in a set of values. In this scenario, the population standard deviation (sigma = 15) indicates how much individual IQ scores of data scientists deviate from the mean. Understanding standard deviation is essential for calculating the sample size, as it influences the width of the confidence interval and the required sample size for a given margin of error.

Recommended video:

Guided course

08:45

Calculating Standard Deviation

Related Practice

Textbook Question

In Exercises 1–4, refer to the accompanying screen display that results from a simple random sample of times (minutes) between eruptions of the Old Faithful geyser. The confidence level of 95% was used.

Interpreting a Confidence Interval The results in the screen display are based on a 95% confidence level. Write a statement that correctly interprets the confidence interval.

Textbook Question

Degrees of Freedom

b. Find the critical value ta/2 corresponding to a 95% confidence level.

views

Textbook Question

Second-Hand Smoke Refer to Data Set 15 “Passive and Active Smoke” and construct a 95% confidence interval estimates of the mean cotinine level in each of three samples: (1) people who smoke; (2) people who don’t smoke but are exposed to tobacco smoke at home or work; (3) people who don’t smoke and are not exposed to smoke. Measuring cotinine in people’s blood is the most reliable way to determine exposure to nicotine. What do the confidence intervals suggest about the effects of smoking and second-hand smoke?

Textbook Question

Heights of Presidents Refer to Data Set 22 “Presidents” in Appendix B. Treat the data as a sample and find the proportion of presidents who were taller than their opponents. Use that result to construct a 95% confidence interval estimate of the population percentage. Based on the result, does it appear that greater height is an advantage for presidential candidates? Why or why not?

Textbook Question

Finding Critical Values

In Exercises 5–8, find the critical value z=a/2 that corresponds to the given confidence level.

99.5%

Textbook Question

In Exercises 5–8, (a) identify the critical value ta/2 used for finding the margin of error, (b) find the margin of error, (c) find the confidence interval estimate of u, and (d) write a brief statement that interprets the confidence interval.

Birth Weights Here are summary statistics for randomly selected weights of newborn girls: n=36, x=3150.0g, s=695.5g (based on Data Set 6 “Births” in Appendix B). Use a confidence level of 95%.