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Chapter 8: Problem 103

The average height of young adult males has a normal distribution with standard deviation of 2.5 inches. You want to estimate the mean height of students at your college or university to within one inch with 93% confidence. How many male students must you measure?

Short Answer

Expert verified
You need to measure 21 male students.

Step by step solution

01

Understand the Given Information

We are provided the standard deviation \( \sigma = 2.5 \) inches. We need to estimate the mean height to within one inch with 93% confidence. The margin of error (E) is 1 inch.
02

Determine the Z-Score for 93% Confidence

The confidence level is 93%, which means there is 7% in the tails. For a two-tailed confidence interval, 3.5% in each tail. The Z-score corresponding to a cumulative probability of 0.965 in standard normal distribution (1 - 0.035 in the upper tail) is approximately 1.81.
03

Use the Sample Size Formula

The formula for sample size when estimating a population mean is \[ n = \left( \frac{Z \times \sigma}{E} \right)^2 \]. Substituting the known values: \( Z = 1.81 \), \( \sigma = 2.5 \), and \( E = 1 \), we have \[ n = \left( \frac{1.81 \times 2.5}{1} \right)^2 \].
04

Calculate the Sample Size

Calculate \( \frac{1.81 \times 2.5}{1} = 4.525 \). Now, square this value to find \( n \): \( n = (4.525)^2 \approx 20.475625 \).
05

Round Up to Nearest Whole Number

Since the sample size must be a whole number, and we always round up to ensure precision in confidence interval width, we round 20.475625 up to 21.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Margin of Error
When conducting surveys or experiments, the margin of error (often abbreviated as E) reflects how much the estimated mean of a sample might differ from the actual population mean. It provides a range of error that is acceptable in our estimates. In scientific research or statistical analysis, a smaller margin of error provides greater precision in our findings.

- In the context of sample size determination, the margin of error is a crucial factor. It sets the maximum allowable difference between the sample mean and the population mean. - If a smaller margin of error is needed, a larger sample size is required to increase the accuracy of the estimate. - In the exercise, the margin of error is 1 inch. This means that the mean height estimated from the sample could differ by up to 1 inch from the true population mean height.
Confidence Interval
A confidence interval is a range of values that you can be reasonably sure contains the true population parameter. In simpler terms, it reflects the degree of certainty (or uncertainty) about our estimates.

- The confidence interval is determined by the confidence level, which is given as a percentage. For example, a 93% confidence level suggests that if you were to take multiple samples and compute an interval each time, approximately 93 out of 100 intervals will contain the actual population mean. - Boundaries of the confidence interval depend on the standard deviation, margin of error, and Z-score. - In this context, a 93% confidence interval implies that our sample will reliably represent the population provided we base our estimate within this range.
Standard Deviation
Standard deviation is a measure of the amount of variation or dispersion in a set of values. It is an indication of how much individual data points differ from the mean of the dataset.

- A smaller standard deviation indicates that data points tend to be close to the mean, while a larger standard deviation suggests more variability. - In sample size determination, knowing the standard deviation helps assess the variability in the population. It plays a significant role in calculating the sample size needed for estimating the population mean accurately. - In this problem, the standard deviation of the height of young adult males is given as 2.5 inches, which represents the average difference from the typical height around the mean.
Z-Score
A Z-score is a statistical measurement describing a value's relation to the mean of a group of values. It is expressed in terms of standard deviations from the mean. In the realm of confidence intervals and sample size determination, Z-scores are used to translate the confidence level into a specific value for use in calculations.

- Z-scores reflect how many standard deviations an element is from the mean. They are critical in determining the width of a confidence interval. - In the context of a confidence interval, a Z-score corresponds to the number of standard deviations needed to achieve a certain level of confidence. - Given a 93% confidence level in this exercise, the appropriate Z-score was determined to be around 1.81. This Z-score helps ensure that we capture the true population mean within our confidence interval 93% of the time.

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Most popular questions from this chapter

Use the following information to answer the next ten exercises: A sample of 20 heads of lettuce was selected. Assume that the population distribution of head weight is normal. The weight of each head of lettuce was then recorded. The mean weight was 2.2 pounds with a standard deviation of 0.1 pounds. The population standard deviation is known to be 0.2 pounds. In words, define the random variable \(\overline{X}\) .

In a recent Zogby International Poll, nine of 48 respondents rated the likelihood of a terrorist attack in their community as likely or very likely. Use the plus four鈥� method to create a 97% confidence interval for the proportion of American adults who believe that a terrorist attack in their community is likely or very likely. Explain what this confidence interval means in the context of the problem.

Use the following information to answer the next seven exercises: The U.S. Census Bureau conducts a study to determine the time needed to complete the short form. The Bureau surveys 200 people. The sample mean is 8.2 minutes. There is a known standard deviation of 2.2 minutes. The population distribution is assumed to be normal. In words, define the random variables \(X\) and \(\overline{X}\) .

Use the following information to answer the next five exercises. Suppose the marketing company did do a survey. They randomly surveyed 200 households and found that in 120 of them, the woman made the majority of the purchasing decisions. We are interested in the population of households where women make the majority of the purchasing decisions. Identify the following: a. \(x=\) _____ b. \(n=\) ______ c. \(p^{\prime}=\) _____

Use the following information to answer the next 14 exercises: The mean age for all Foothill College students for a recent Fall term was 33.2. The population standard deviation has been pretty consistent at 15. Suppose that twenty-five Winter students were randomly selected. The mean age for the sample was 30.4. We are interested in the true mean age for Winter Foothill College students. Let X = the age of a Winter Foothill College student. As a result of your answer to Exercise 8.26, state the exact distribution to use when calculating the confidence interval.
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