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Chapter 7: Q. 7.29 (page 300)

Does the sample size have an effect on the standard deviation of all possible sample means? Explain your answer.

Short Answer

Expert verified

Yes, the sample size has an effect on the standard deviation of all possible sample means.

Step by step solution

01

Step 1. Given Information

We need to identify whether the sample size has an effect on the standard deviation of all possible sample means.

02

Step 2. Explanation

For samples of size n,the standard deviation of the variable $\bar{x}$ equals the standard deviation of the variable under consideration divided by the square root of the sample size. That is, $σ_{\bar{x}} = \frac{σ}{\sqrt{n}}$ .

So, it can be seen that the standard deviation of all possible sample means depends on the sample size n.

And it can be seen that as the sample size increase the standard deviation of $\bar{x}$ decreases.

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Most popular questions from this chapter

Officer Salaries. The following table gives the monthly salaries (in $\(1000$ ) of the six officers of a company.

a. Calculate the population mean monthly salary, $μ$

There are $15$ possible samples of size $4$ from the population of six officers. They are listed in the first column of the following table.

b. Complete the second and third columns of the table.

c. Complete the dot plot for the sampling distribution of the sample mean for samples of size $4$ Locate the population means on the graph.

d. Obtain the probability that the mean salary of a random sample of four officers will be within 1 (i.e., $\) 1000$ ) of the population mean.

Repeat parts (b)-(e) of Exercise 7.11 for samples of size4.

A variable of a population is normally distributed with mean $μ$ and standard deviation $σ$ . For samples of size $n$ , fill in the blanks. Justify your answers.

a. Approximately $68 %$ of all possible samples have means that lie within of the population mean, $μ$

b. Approximately $95 %$ of all possible samples have means that lie within of the population mean, $μ$

c. Approximately $99.7 %$ of all possible samples have means that lie within of the population mean, $μ$

d. $100 (1 - α) %$ of all possible samples have means that lie within $_$ of the population mean, $μ$ (Hint: Draw a graph for the distribution of $x$ , and determine the $z -$ scores dividing the area under the normal curve into a middle $1 - α$ area and two outside areas of $α / 2$

Refer to Fig. $7.6$ on page 306 .

a. Why are the four graphs in Fig. 7.6(a) all centered at the same place?

b. Why does the spread of the graphs diminish with increasing sample size? How does this result affect the sampling error when you estimate a population mean, $μ$ by a sample mean, $\tilde{x}$ ?

c. Why are the graphs in Fig. 7.6(a) bell shaped?

d. Why do the graphs in Figs. $7.6 (b)$ and (c) become bell shaped as the sample size increases?

Alcohol consumption on college and university campuses has gained attention because undergraduate students drink significantly more than young adults who are not students. Researchers I. Balodis et al. studied binge drinking in undergraduates in the article "Binge Drinking in Undergraduates: Relationships with Gender, Drinking Behaviors, Impulsivity, and the Perceived Effects of Alcohol". The researchers found that students who are binge drinkers drink many times a month with the span of each outing having a mean of 4.9 hours and a standard deviation of 1.1 hours.

Part (a): For samples of size 40, find the mean and standard deviation of all possible sample mean spans of binge drinking episodes. Interpret your results in words.

Part (b): Repeat part (a) with $n = 120$ .

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