Sampling Distributions

Question 1

Characteristics of a discrete probability distribution

Answer 1

• list of outcomes is exhaustive
• outcomes are mutually exclusive
• probabilities sum to 1

Question 2

Characteristics of continuous probability distributions

Answer 2

• have PDFs
• total area under curve = 1

Question 3

Sampling error

Answer 3

Different samples yield different values for the same statistics

Question 4

Sampling distribution

Answer 4

A probability distribution of a sample statistic resulting from repeated sampling

Question 5

The standard deviation of the sampling distribution is usually called the _______ of the mean.

Answer 5

standard error

Question 6

_______ concerns the center of the sampling distribution. A statistic used to estimate a parameter is an _______ estimator if the mean of its sampling distribution is equal to the true value of the parameter being estimated.

Answer 6

Bias, unbiased

Question 7

The _______ of a statistic is described by the spread of its sampling distribution.

Answer 7

variability

Question 8

This _______ is determined by the sampling design and the sample size n. Statistics from larger random samples have smaller _______.

Answer 8

spread, spreads

Question 9

To reduce bias, use _______.

Answer 9

random sampling

Question 10

When we start with a list of the entire population, simple random sampling (SRS) produces _______ estimates—the values of a statistic computed from an SRS neither consistently overestimate nor consistently underestimate the true parameter value.

Answer 10

unbiased

Question 11

To reduce the variability of a statistic from an SRS, use a _________. You can make the variability as small as you want by taking a _______ enough sample.

Answer 11

larger sample, large

Question 12

T or F
Large populations do not require large samples.

Answer 12

True, the variability of a statistic from a random sample depends little on the size of the population, as long as the population is at least 20 times larger than the sample.

Question 13

The _______ is a numerical measure of a statistic’s precision and is a function of the spread of the sampling distribution. It sets bounds on the size of the likely error when the statistic is used as the estimator of a parameter.

Answer 13

margin of error

Question 14

Population distribution

Answer 14

The population distribution of a variable is the distribution of its values for all members of the population. The population distribution is also the probability distribution of the variable when we choose one individual at random from the population.

Question 15

Facts about sample means

Answer 15

Sample means are less variable than individual observations.
Sample means are centered on the population mean.
For large n, the distribution of sample means is close to Normal.

Question 16

When n is large, _______ the states that the sampling distribution of the sample mean is approximately Normal.

Answer 16

central limit theorem

Question 17

You take an SRS of size 25 from a population with mean 215 and standard deviation 10. According to the central limit theorem, what is the approximate sampling distribution of the sample mean? Use the 95 part of the 68–95–99.7 rule to determine the margin of error.

Answer 17

About 95% of the time x-bar should be between 21 and 29.

Question 18

What’s wrong? For each of the following statements, explain what is wrong and why.

If the population standard deviation is 20, then the standard deviation of x-bar for an SRS of 10 observations is 20/10=2.

Answer 18

It will be 20/sqrt(10)

Question 19

What’s wrong cont.

When taking SRSs from a population, larger sample sizes will result in larger standard deviations of x-bar.

Answer 19

Larger samples result in smaller deviations.

Question 20

What’s wrong cont.

For an SRS from a population, both the mean and the standard deviation of x-bar depend on the sample size n.

Answer 20

Only the sd is dependent.

Question 21

What’s wrong cont.

The larger the population, the bigger the sample size n needs to be for a desired standard deviation of x-bar.

Answer 21

Sample size isn’t dependent on population size.

Question 22

Cholesterol levels of teenagers. A study of the health of teenagers plans to measure the blood cholesterol level of an SRS of 13- to 16-year-olds. The researchers will report the mean x-bar from their sample as an estimate of the mean cholesterol level μ in this population.

Explain to someone who knows no statistics what it means to say that x-bar is an “unbiased” estimator of μ.

Answer 22

The sample mean is not systematically higher or lower than the population mean.

Question 23

Cholesterol levels of teenagers. A study of the health of teenagers plans to measure the blood cholesterol level of an SRS of 13- to 16-year-olds. The researchers will report the mean x-bar from their sample as an estimate of the mean cholesterol level μ in this population.

Explain to someone who knows no statistics why a large sample gives more trustworthy results than a small sample.

Answer 23

A larger sample provides more data, allowing sample mean to be more accurate to population mean.

Question 24

In 2019, the Federal Aviation Administration (FAA) updated its standard average passenger weight to be based on data from U.S. government health agency surveys.12 It specified this average weight, which includes clothing, as 189 pounds in the summer (195 in the winter). These health agency surveys can also be used to determine the standard deviation, which we’ll assume is 47 pounds. Weights are not Normally distributed, especially when the population includes both men and women, but they are not very non-Normal. A commuter plane carries 25 passengers. What is the approximate probability that, in the winter, the total weight of the passengers exceeds 5225 pounds? (Hint: To apply the central limit theorem, restate the problem in terms of the mean weight.)

Answer 24

0.0681

Question 25

A poll of 1240 college students asked whether or not they used the Internet to find their current place to live. There were 904 students who answered Yes; the other 336 answered No.

A. What is the sample size n?

B. Choose one of the two possible outcomes to define the random variable X. Give a reason for your choice.

c. What is the value of the count X?

D. Find the sample proportion.

Answer 25

A. 1240

B. Yes, because this is the answer I assume they’re most interested in. (no works too)

C. Yes: 904
No: 336

D. 904/1240
336/1240

Question 26

The distribution of the count X of successes in the binomial setting is called the _______ with parameters n and p. The parameter n is the number of observations, and p is the probability of a success on any one observation. The possible values of X are the whole numbers from 0 to n. As an abbreviation, we say that the distribution of X is B(n, p).

Answer 26

binomial distribution

Question 27

The _______; is an unbiased estimator of the population proportion p.

Answer 27

sample proportion of successes p=x/n

Question 28

The _______ improves the accuracy of the Normal approximations.

Answer 28

continuity correction