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Chapter 7: Q.7.23 (page 296)

America's Richest. Explain what the dotplots in part (c) of exercise 7.17-7.22 illustrate about the impact of increasing sample size on sampling error.

Short Answer

Expert verified

The impact of increasing sample size on sampling error is that The sample size is raised, the sampling error decreases.

Step by step solution

01

Given Information

We have to illustrate about the impact of increasing sample size on sampling error.

02

Explanation

Sampling error arises when only a portion of the population is utilised to estimate population parameters and draw inferences about the population.

For the supplied population, use MINITAB to create dotplots for samples of size $1, 2, 3, 4, 5 a n d 6$ .

MINITAB output is given below:

The population mean height of six persons $46.5$ billion. The MINITAB result clearly shows that as the sample size is raised, the sampling error decreases.

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Most popular questions from this chapter

America's Riches. Each year, Forbes magazine publishes a list of the richest people in the United States. As of September l6, 2013, the six richest Americans and their wealth (to the neatest billion dollars) are as shown in the following table. Consider these six people a population of interest.

(a) For sample size of $5$ construct a table similar to table 7.2 on page293.(There are 6 possible sample) of size $5$

(b) For a random sample of size $5$ determine the probability that themean wealth of the two people obtained will be within $3$ (i.e, $3$ billion) of the population mean. interpret your result in terms of percentages.

7.46 Young Adults at Risk. Research by R. Pyhala et al. shows that young adults who were born prematurely with very low birth weights (below $1500$ grams) have higher blood pressure than those born at term. The study can be found in the article. "Blood Pressure Responses to Physiological Stress in Young Adults with Very Low Birth Weight" (Pediatrics, Vol. 123, No, 2, pp. $731 - 734$ ). The researchers found that systolic blood pressures, of young adults who were born prematurely with very low birth weights have mean $120.7 mmHg$ and standard deviation $13.8 mmHg$ .
a. Identify the population and variable.
b. For samples of $30$ young adults who were born prematurely with very low birth weights, find the mean and standard deviation of all possible sample mean systolic blood pressures. Interpret your results in words.
c. Repeat part (b) for samples of size $90$ .

A statistic is said to be an unbiased estimator of a parameter if the mean of all its possible values equals the parameter; otherwise, it is said to be a biased estimator. An unbiased estimator yields, on average, the correct value of the parameter, whereas a biased estimator does not.

Part (a): Is the sample mean an unbiased estimator of the population mean? Explain your answer.

Part (b): Is the sample median an unbiased estimator of the population mean? Explain your answer.

The winner of the 2012-2013 National Basketball Association (NBA) championship was the Miami Heat. One possible starting lineup for that team is as follows.

a. Determine the population mean height, $μ$ , of the five players:

b. Consider samples of size $2$ without replacement. Use your answer to Exercise $7.11 (b)$ on page $295$ and Definition $3.11$ on page $140$ to find the mean, $μ_{r}$ , of the variable $\hat{x}$ .

c. Find $μ_{x^{*}}$ using only the result of part (a).

In Example 7.5, we used the definition of the standard deviation of a variable to obtain the standard deviation of the heights of the five starting players on a men's basketball team and also the standard deviation of $x$ for samples of sizes 1,2,3,4,5.The results are summarized in Table 7.6on page 298. Because the sampling is without replacement from a finite population, Equation (7.1) can also be used to obtain $σ_{x}$ .

Part (a): Apply Equation (7.1) to compute $σ_{x}$ for sample sizes of 1,2,3,4,5. Compare your answers with those in Table 7.6.

Part (b): Use the simpler formula, Equation (7.2) to compute $σ_{x}$ for samples of sizes 1,2,3,4,5.Compare your answers with those in Table 7.6. Why does Equation (7.2)generally yield such poor approximations to the true values?

Part (c): What percentages of the population size are samples of sizes 1,2,3,4,5.

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