Deck 10: Sampling Distributions

Question 1

A sample of size 50 is to be taken from an infinite population whose mean and standard deviation are 52 and 20, respectively. The probability that the sample mean will be larger than 49 is:

Answer 1

C

Question 2

If all possible samples of size n are drawn from an infinite population with mean $\mu$ and standard deviation $\sigma$ , then the standard error of the sample mean is inversely proportional to: $\begin{array}{|l|l|}\hline\text { A. } & \mu . \\\hline \text { B. } & \sigma . \\\hline \text { C. } & n \\\hline \text { D. } & \sqrt{n} \\\hline\end{array}$

Answer 2

D

Question 3

Given an infinite population with a mean of 75 and a standard deviation of 12, the probability that the mean of a sample of 36 observations, taken at random from this population, exceeds 78 is:

Answer 3

B

Question 4

The finite population correction factor should not be used when:

Question 5

If all possible samples of size n are drawn from a population, the probability distribution of the sample mean is called the:

Question 6

As a general rule in computing the standard error of the sample mean, the finite population correction factor is used only if the:

Question 7

A normally distributed population with 200 elements has a mean of 60 and a standard deviation of 10. The probability that the mean of a sample of 25 elements taken from this population will be smaller than 56 is:

Question 8

Random samples of size 64 are taken from an infinite population whose mean is 160 and standard deviation is 32. The mean and standard error of the sample mean, respectively, are:

Question 9

If a random sample of size n is drawn from a normal population, then the sampling distribution of the sample mean will be:

Question 10

The standard deviation of the sampling distribution of the sample mean is also called the:

Question 11

Consider an infinite population with a mean of 160 and a standard deviation of 25. A random sample of size 64 is taken from this population. The standard deviation of the sample mean equals:

Question 12

If all possible samples of size n are drawn from an infinite population with a mean of 15 and a standard deviation of 5, then the standard error of the sample mean equals 1.0 only for samples of size:

Question 13

A population that consists of 250 items has a mean of 37 and a standard deviation of 13. A sample of size 5 is taken at random from this population. The standard error of the sample mean equals (up to three decimal places):

Question 14

An infinite population has a mean of 33 and a standard deviation of 6. A sample of 100 observations is to be taken at random from this population. The probability that the sample mean will be between 34.5 and 36.1 is:

Question 15

A sample of size n is selected at random from an infinite population. As n increases, which of the following statements is true? $\begin{array}{|l|l|}\hline A&\text {The population standard deviation decreases. }\\\hline B&\text {The standard error of the sample mean decreases. }\\\hline C&\text { The population standard deviation increases.}\\\hline D&\text {The standard error of the sample mean increases. }\\\hline \end{array}$

Question 16

A sample of size 35 is selected at random from a finite population. If the finite population correction factor is 0.5, then (rounded to the nearest integer) the population size is:

Question 17

Random samples of size 81 are taken from an infinite population whose mean and standard deviation are 45 and 9, respectively. The mean and standard error of the sampling distribution of the sample mean are:

Question 18

The central limit theorem states that if a random sample of size n is drawn from a population, then the sampling distribution of the sample mean :

Question 19

Which of the following best describes the Central Limit Theorem?

Question 20

The expected value of the sampling distribution of the sample mean $\bar { X }$ equals the population mean $\mu$ : $\begin{array}{|l|l|}\hline A&\text { when the population is normally distributed.}\\\hline B&\text {when the population is symmetric. }\\\hline C&\text {when the population size \( N>30 \). }\\\hline D&\text { for all populations.}\\\hline \end{array}$

Question 21

A sample of size n is selected at random from an infinite population. As n increases the standard error of the sample mean decreases.

Question 22

When a great many simple random samples of size n are drawn from a population that is normally distributed, the sampling distribution of the sample means will be normal, regardless of sample size n.

Question 23

If all possible samples of size n are drawn from a population, the probability distribution of the sample mean $\bar { X }$ is referred to as the normal distribution.

Question 24

The sampling distribution of the sample proportion has mean p and variance pq/n.

Question 25

If the daily demand for boxes of mineral water at a supermarket is normally distributed with a mean of 47.6 boxes and a standard deviation of 5.8 boxes, what is the probability that the average demand for a sample of 10 supermarkets will be less than 50 boxes in a given day?

Question 26

The sampling distribution of the sample proportion is approximately normal provided that np ≥ 5 and nq ≥ 5

Question 27

It is know that 40% of voters in a certain electorate are in favour of a particular candidate. If a sample of size 30 is taken, what is the probability that less than 35% are in favour of this political candidate?

Question 28

Which of the following best describes the sampling distribution of the sample proportion?

Question 29

The standard deviation of a sampled population is also called the standard error of the sample mean.

Question 30

Suppose that the time needed to complete a final exam is normally distributed with a mean of 85 minutes and a standard deviation of 18 minutes.
a. What is the probability that the total time taken by a group of 100 students will not exceed 8200 minutes?
b. What assumption did you have to make in your computations in part (a)?

Question 31

A manufacturing company is concerned about the number of defective items produced by their assembly line. In the past they have had 5% of their products produced defectively. They take a random sample of 35 products. What is the probability that more than 5 products in the sample are defective?

Question 32

A researcher conducted a survey on a university campus for a sample of 64 third-year students and reported that third-year students read an average of 3.12 books in the prior academic semester, with a standard deviation of 2.15 books. Determine the probability that the sample mean is:
a. less than 3.45.
b. between 3.38 and 3.58.
c. above 2.94.

Question 33

Consider an infinite population with a mean of 100 and a standard deviation of 20. A random sample of size 50 is taken from this population. The standard deviation of the sample mean equals 3.2.

Question 34

The sampling distribution of the sample proportion is always normally distributed.

Question 35

If it is known that the population proportion of car parks in a particular city that offer early discount fees is 25%, which of the following best describes the mean and standard deviation of the sampling distribution of the sample proportion of car parks that offer early discount fees for samples taken of size 30?

Question 36

The standard error of the mean is the standard deviation of the sampling distribution of $\bar { X }$ .

Question 37

If all possible samples of size n are drawn from an infinite population with a mean of 60 and a standard deviation of 8, then the standard error of the sample mean equals 1.0 only for samples of size 64.

Question 38

A sample of size 70 is selected at random from a finite population. If the finite population correction factor is 0.808, the population size (rounded to the nearest integer) must be 200.

Question 39

The central limit theorem is basic to the concept of statistical inference, because it permits us to draw conclusions about the population based strictly on sample data, and without having any knowledge about the distribution of the underlying population.

Question 40

An infinite population has a mean of 120 and a standard deviation of 44. A sample of 100 observations is to be selected at random from the population.
a. What is the expected value of the sample mean?
b. What is the standard deviation of the sample mean?
c. What is the shape of the sampling distribution of the sample mean?
d. What does the sampling distribution of the sample mean show?

Question 41

A sample of 30 observations is drawn from a normal population with mean of 750 and a standard deviation of 300. Suppose the population size is 600.
a. Find the expected value of the sample mean .
b. Find the standard error of the sample mean .
c. Find P( > 790).
d. Find P( < 650).
e. Find P(760 < < 810).

Question 42

The amount of time spent by Australian adults playing sports per day is normally distributed, with a mean of 4 hours and standard deviation of 1.25 hours. Use this information to answer the following question(s).
Find the probability that a randomly selected Australian adult plays sport for more than 5 hours per day.

Question 43

The heights of 9-year-old children are normally distributed, with a mean of 123 cm and a standard deviation of 10 cm.
a. Find the probability that one randomly selected 9-year-old child is taller than 125 cm.
b. Find the probability that three randomly selected 9-year-old children are taller than 125 cm.
c. Find the probability that the mean height of three randomly selected 9-year-old children is greater than 125 cm.

Question 44

The heights of women in Australia are normally distributed, with a mean of 165 centimetres and a standard deviation of 10 centimetres. What is the probability that the mean height of a random sample of 30 women is smaller than 162 centimetres?

Question 45

The amount of time spent by Australian adults playing sports per day is normally distributed, with a mean of 4 hours and standard deviation of 1.25 hours. Find the probability that if four Australian adults are randomly selected, their average number of hours spent playing sport is more than 5 hours per day.

Question 46

A sample of size 400 is drawn from a population whose mean and variance are 5000 and 10 000, respectively. Find the following probabilities.
a. P( < 4,990).
b. P(4995 < < 5010).
c. P( = 5000).

Question 47

The amount of time spent by Australian adults playing sports per day is normally distributed, with a mean of 4 hours and standard deviation of 1.25 hours. Find the probability that if four Australian adults are randomly selected, all four play sport for more than 5 hours per day.

Question 48

An auditor knows from past history that the average accounts receivable for a company is $521.72, with a standard deviation of $584.64. If the auditor takes a simple random sample of 100 accounts, what is the probability that the mean of the sample will be within $120 of the population mean?

Question 49

The heights of women in Australia are normally distributed, with a mean of 165 centimetres and a standard deviation of 10 centimetres. If the population of women's heights were not normally distributed, which, if any, of the following questions could you answer?

A) What is the probability that a randomly selected woman is taller than 160 cm?
B) A random sample of five women is selected. What is the probability that the sample mean is greater than 160 cm?
C) What is the probability that the mean height of a random sample of 75 women is greater than 160 cm?

Question 50

The heights of women in Australia are normally distributed, with a mean of 165 centimetres and a standard deviation of 10 centimetres. Use this information to answer the following question(s).
What is the probability that a randomly selected woman is shorter than 162 centimetres?

Question 51

Suppose that the average annual income of a lawyer is $150 000 with a standard deviation of $40 000. Assume that the income distribution is normal.
a. What is the probability that the average annual income of a sample of 5 lawyers is more than $120 000?
b. What is the probability that the average annual income of a sample of 15 lawyers is more than $120 000?

Question 52

The heights of women in Australia are normally distributed, with a mean of 165 centimetres and a variance of 100 centimetres². A random sample of five women is selected. What is the probability that the sample mean is greater than 162 centimetres?

Question 53

Assume that the time needed by a worker to perform a maintenance operation is normally distributed, with a mean of 60 minutes and a standard deviation of 6 minutes. What is the probability that the average time needed by a sample of 5 workers to perform the maintenance is between 63 and 68 minutes?

Question 54

A sample of 50 observations is drawn at random from a normal population whose mean and standard deviation are 75 and 6, respectively.
a. What does the central limit theorem say about the sampling distribution of the sample mean? Why?
b. Find the mean and standard error of the sampling distribution of the sample mean.
c. Find P( > 73).
d. Find P( < 74).

Question 55

The following table gives the number of pets owned for a population of four families. $\begin{array} { | l | c | c | c | c | } \hline \text { Family } & \text { A } & \text { B } & \text { C } & \text { D } \\\hline \text { Number of pets owned } & 2 & 1 & 4 & 3 \\\hline\end{array}$ Find the mean and standard deviation for the population.

Question 56

An infinite population has a mean of 100 and a standard deviation of 20. Suppose that the population is not normally distributed. What does the central limit theorem say about the sampling distribution of the mean if samples of size 64 are drawn at random from this population?

Question 57

A sample of size 35 is taken from a normal population, with mean of 65 and standard deviation of 9.3. Describe the sampling distribution of the sample mean.

Question 58

If a random variable X is not known to be normally distributed, has a mean of size 8 and a variance of size 1.5, describe the sampling distribution of the sample mean for samples of size 30.

Question 59

In order to estimate the mean salary for a population of 500 employees, the managing director of a certain company selected at random a sample of 40 employees.
a. Would you use the finite population correction factor in calculating the standard error of the sample mean? Explain.
b. If the population standard deviation is $800, compute the standard error both with and without using the finite population correction factor.
c. What is the probability that the sample mean salary of the employees will be within ±$200 of the population mean salary?

Question 60

In a given year, the average annual salary of an Rugby player was $205 000, with a standard deviation of $24 500. If a simple random sample of 50 players is taken, what is the probability that the sample mean will be less than $210 000?

Question 61

A video rental store wants to know what proportion of its customers are under 21 years old. A simple random sample of 500 customers is taken, and 350 of them are under 21. Assume that the true population proportion of customers aged under 21 is 0.68.
Describe the sampling distribution of proportion of customers who are under age 21.

Question 62

A local newspaper sells an average of 2100 papers per day, with a standard deviation of 500 papers. Consider a sample of 60 days of operation.
a. What is the shape of the sampling distribution of the sample mean number of papers sold per day? Why?
b. Find the expected value and the standard error of the sample mean.
c. What is the probability that the sample mean will be between 2000 and 2300 papers?

Question 63

A video rental store wants to know what proportion of its customers are under 21 years old. A simple random sample of 500 customers is taken, and 350 of them are under 21. Assume that the true population proportion of customers aged under 21 is 0.68. If another simple random sample of size 500 is taken, what is the probability that more than 70% of customers are under 21 years old?

Question 64

The following table gives the number of pets owned for a population of four families. List all possible samples of two families that can be selected without replacement from this population, and compute the sample mean for each sample.

Question 65

Using the following sampling distribution, directly recalculate the mean and standard deviation of .

Question 66

The following table gives the number of pets owned for a population of four families. $\begin{array} { | l | c | c | c | c | } \hline \text { Family } & \text { A } & \text { B } & \text { C } & \text { D } \\\hline \text { Number of pets owned } & 2 & 1 & 4 & 3 \\\hline\end{array}$ Samples of size 2 will be drawn at random from the population. Use your answers to the previous question to calculate the mean and the standard deviation of the sampling distribution of the sample means.

Question 67

A video rental store wants to know what proportion of its customers are under 21 years old. A simple random sample of 500 customers is taken, and 375 of them are under 21. Assume that the true population proportion of customers aged under 21 is 0.68.
What is the probability that the sample proportion will be within 0.03 of the true proportion of customers who are aged under 21?

Question 68

The following table gives the number of pets owned for a population of four families. Find the sampling distribution of .