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Chapter 7: Q. 18 (page 312)

The Athletic Coping Skills Inventory (ACSI) is a test to measure psychological skills believed to influence athletic performance. Researchers E. Estanol er al. studied the relationship between ACSI scores and eating disorders in dancers in the article "Mental Skills as Protective Attributes Against Eating Disorder Risk in Dancers. The study found that dancers ACSI scores are normally distributed with mean \(49.34\) points and standard deviation \(10.14\) points.

a. Use the technology of your choice to simulate \(1000\) samples of four dancers ACSI scores each.

b. Find the sample mean of each of the \(1000\) samples obtained in part (a).

c. Obtain the mean, the standard deviation and a histogram of the \(1000\) sample means.

d. Theoretically what are the mean, standard deviation and distribution of all possible sample means for samples of size \(4\)?

e. Compare your answers from parts (c) and (d).

Short Answer

Expert verified

Part a. The samples will be like

Part b. The mean will be \(\mu=[49.3615, 49.1953, 48.7853, 49.1773]\)

Part c. The standard deviation and mean will be

\(\mu_{bar{x}}=49.6020\)

\(\sigma_{bar{x}}=\frac{10.14}{\sqrt{4}}=5.07\)

Part d. \(\mu_{\bar{x}}=\mu=49.34

\(\sigma_{\bar{x}}=\frac{\sigma}{\sqrt{n}}=\frac{10.14}{\sqrt{4}}=5.07\)

Part e. We can see that the mean part (c) is slightly lower than the theoretical mean of part (d), whereas the standard deviation of part (c) is slightly higher than part (d).

However, we do note that the mean and standard deviation of part (c) and (d) are close in value.

Step by step solution

01

Part a. Step 1. Given information

The sample size, mean and standard deviation is given.

\(\mu=mean=49.34\)

\(\sigma=standard deviation=10.14\)

\(n=sample size=4\)

02

Part a. Step 2. Calculation

So, according to the question I have used MATLAB to generate the samples.

We use the function \(鈥渘orminv (rand (1000,4), 49.34, 10.14)鈥漒).

After using the above function in MATLAB, we will get the \(1000\) sample for \(4\) dancers.

Program:

Query:

  • First, we have defined the number of samples.
  • Then using function 鈥渘orminv鈥� to find out the samples for dancers.
03

Part b. Step 1. Calculation

So, according to the question I have used MATLAB to generate the samples.

We use the function \(鈥渘orminv (rand (1000,4), 49.34, 10.14)鈥漒).

Then calculate the mean

\(\mu=\frac{\sum_{i-1}^{1000}x_{i}}{n}\)

After solving the equation, we will get

\(\mu=[49.3615, 49.1953, 48.7853, 49.1773]\)

Program:

Query:

  • First, we have defined the number of samples.
  • Then using function 鈥渘orminv鈥� to find out the samples for dancers.
  • Then calculate the mean using function 鈥渕ean鈥�.
04

Part c. Step 1. Calculation

So, according to the question I have used MATLAB to generate the samples.

We use the function \(鈥渘orminv (rand (1000,4), 49.34, 10.14)鈥漒).

Then calculate the mean

\(\mu=\frac{\sum_{i-1}^{1000}x_{i}}{n}\)

After solving the equation, we will get

\(\mu=[49.3615, 49.1953, 48.7853, 49.1773]\)

Calculate the mean of means

\(\mu_{bar{x}}=\bar{\mu}\)

\(\mu_{bar{x}}=49.6020\)

and standard deviation

\(\sigma_{\bar{x}}=\frac{10.14}{\sqrt{4}}=5.07\)

Program:

Query:

  • First, we have defined the number of samples.
  • Then using function 鈥渘orminv鈥� to find out the samples for dancers.
  • Then calculate the mean using function 鈥渕ean鈥�.
  • Calculate the mean of means and standard deviation.
  • Sketch a histogram.
05

Part d. Step 1. Explanation

The mean of the sampling distribution is equal to the population mean.

\(\mu_{\bar{x}}=\mu=49.34\)

The standard deviation of the sampling distribution is equal to the population standard deviation divide by the square root of the sample size.

\(\sigma_{\bar{x}}=\frac{10.14}{\sqrt{4}}=5.07\)

06

Part e. Step 1. Explanation

The mean of the sampling distribution is equal to the population mean.

\(\mu_{\bar{x}}=\mu=49.6020\)

\(\mu_{\bar{x}}=\mu=49.34\)

The standard deviation of the sampling distribution is equal to the population standard deviation divide by the square root of the sample size.

\(\sigma_{\bar{x}}=\frac{10.14}{\sqrt{4}}=5.07\)

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Most popular questions from this chapter

Ethanol Railroad Tariffs. An ethanol railroad tariff is a fee charged for shipments of ethanol on public railroads. The Agricultural Marketing Service publishes tariff rates for railroad-car shipments of ethanol in the Biofuel Transportation Database. Assuming that the standard deviation of such tariff rates is \(1,150, determine the probability that the mean tariff rate of 500randomly selected railroad car shipments of ethanol will be within \)100of the mean tariff rate of all railroad-car shipments of ethanol. Interpret your answer in terms of sampling error.

As reported by the U.S. Census Bureau in Educational Attainment in the United States, the percentage of adults in each state who have completed a bachelor's degree is provided on the Weiss Stats site. Use the technology of you choice to solve the following problems.

Part (a): Obtain the standard deviation of the variable "percentage of adults who have completed a bachelor's degree" for the population of 50 states.

Part (b): Consider simple random samples without replacement from the population of 50 states. Strictly speaking, which is the correct formula for obtaining the standard deviation of the sample mean- Equation (7.1) or Equation (7.2)? Explain your answer.

Part (c): Referring to part (b), obtain R for simple random samples of size 30 by using both formulas. Why does Equation (7.2) provide such a poor estimate of the true value given by Equation (7.1)?

Part (d): Referring to part (b), obtain R for simple random samples of size 2 by using both formulas. Why does Equation (7.2) provide a somewhat reasonable estimate of the true value given by Equation (7.1)?

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,渭

There are 15possible samples of size 4from 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 4Locate 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.

A variable of a population has mean 渭 and standard deviation蟽. that For a large sample size n, answer the following questions.

a. Identify the distribution ofx.

b. Does your answer to part (a) depend on n being large? Explain your answer.

c. Identify the mean and the standard deviation ofx.

d. Does your answer to part (c) depend on the sample size being large? Why or why not?

7.49 Mobile Homes. According to the U.S. Census Bureau publication Manufactured Housing Statistics, the mean price of new mobile homes is \(65,100. Assume a standard deviation of \)7200. Let x~ denote the mean price of a sample of new mobile homes.
a. For samples of size 50, find the mean and standard deviation of x炉. Interpret your results in words.
b. Repeat part (a) with n=100.
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