Deck 11: Random Sampling and Sampling Distributions

Question 1

A sociologist engaged in research on religious attitudes tells her assistant to select a random sample of ten members for interviews from a large suburban church. The sample selected consists of nine women and one man. Upon seeing the uneven distribution of sexes in the sample, the sociologist complains, "This sample can't be random: it's almost all women." Any comments?

Answer 1

Whether or not a sample is random depends on how it was selected, not on its composition; the chance factors involved in random sampling occasionally lead to very atypical samples.

Question 2

Explain on an intuitive basis why the sampling distribution of means for

selected from a rectangular distribution has more cases in the middle than at the extremes.

Answer 2

In order to obtain an extremely low or extremely high mean, both scores would have to be extremely low or extremely high, events that would occur with relatively low frequency; on the other hand, there are many ways sample means toward the center can be obtained (a combination of a very high and a very low score, two moderate scores, etc.), and thus they occur with greater frequency.

Question 3

For a normally distributed population of observations,

and

For a randomly selected sample of size 36, what is the probability that the sample mean
(a) will fall above 91
(b)will fall more than 10 points above the population mean of 100
(c)will differ (in either direction) from the population mean by 10 points or more
(d)What sample mean has such a high value that the probability is .01 of obtaining one as high or higher
(e)Within what limits would the central 95% of all possible sample means fall

Answer 3

(a).96
(b).02
(c).05
(d)111.6
(e)90.2-109.8.

Question 4

A large population of peer ratings of physical attractiveness is approximately normal with

and

We plan to select a random sample of four ratings from this population. What is the probability of obtaining a sample mean
(a) above 6.0
(b)below 4.0
(c)between the population mean and .5 above the mean
(d)no more than .5 away from the population mean (in either direction)
(e)What sample mean has such a low value that the probability is .05 of obtaining one as low or lower?
(f)What are the centrally placed limits such that the probability is .95 that the sample mean will fall within those limits?

Question 5

A widely used IQ test for children is constructed to have a normal distribution with a mean of 100 and a standard deviation of 15 for children in a very, very large national sample. If a child is selected at random from the national sample, what is the probability that his or her IQ score will be
(a) above 90
(b)between 90 and 110
(c)above 150
(d)The probability is .50 that the child's IQ score will be within (how many) IQ points of the mean (on either side)
(e)The probability is .10 that the child's IQ score will fall above what IQ score

Question 6

In statistical work, it is preferable to define a population as

A) a set of objects
B) a set of people
C) a set of observations
D) any of the above

Question 7

A basic aim of statistical inference is to form a conclusion about a

A) sample
B) random sample
C) population
D) random population

Question 8

In the following statement, which alternative is false? The characteristics of a random sample

A) vary from sample to sample
B) vary less with larger samples
C) make statistical inference possible
D) duplicate those of the population if sampling is truly random

Question 9

A random sample is defined by

A) the method of selection
B) how closely it resembles the population
C) the outcome of selection
D) all of the above

Question 10

Which, if any, is not a characteristic of random sampling?

A) whether a sample is random or not cannot be told from inspection of the sample
B) characteristics of a random sample may differ widely from characteristics of its population
C) a sample must be reasonably large to be considered a random sample
D) every element in the population must be given an equal chance for inclusion in the sample

Question 11

The fundamental condition that permits proper statistical inference is

A) random sampling
B) normal distribution of scores
C) knowledge of the values of the parameters of the population
D) having a large sample

Question 12

If the population consists of four scores: 5, 6, 7, and 8, and we are told that a sample of three scores from this population is: 7, 8, 8, we know that

A) sampling was done with replacement
B) sampling was done without replacement
C) sampling was random
D) sampling was not random

Question 13

A sampling distribution is a distribution of

A) scores obtained from samples
B) values of a statistic obtained from samples
C) values of a parameter obtained from samples
D) any of the above

Question 14

Each score in a random sampling distribution of the mean represents

A) a single individual
B) a random data point
C) a standard score
D) a sample mean

Question 15

A particular sampling distribution of the mean is based on means of

A) all possible samples of the same size
B) n samples of the same size
C) all possible samples of all possible sizes
D) n samples of all possible sizes

Question 16

A sampling distribution is

A) a relative frequency distribution
B) a probability distribution
C) both of the above
D) neither of the above

Question 17

The difference between a distribution of sample scores and a sampling distribution is that the latter

A) has characteristics specified by statistical theory
B) is a distribution of sample statistics
C) has a mean and standard deviation which can be known in advance
D) is characterized by all of the above, while the former is not

Question 18

Which is not a characteristic of the random sampling distribution of means?

A) its mean is the same as the mean of the population of scores
B) its standard deviation is greater than that of the population of scores
C) it tends to resemble the normal curve irrespective of the shape of the population of scores
D) its standard deviation changes with variation in sample size

Question 19

The mean of the sampling distribution of means

A) increases as sample size increased
B) changes as the standard deviation of the population of scores is increased
C) changes from
if the population is not normally distributed
D) is unaffected by any of the above

Question 20

The standard deviation of the sampling distribution of means

A) increases as sample size increases
B) increases as the mean of the population increases
C) increases as the standard deviation of the population decreases
D) none of the above is true

Question 21

The standard error of the mean

A) is given in terms of standard units
B) is a standard deviation
C) is larger for larger populations
D) is the average amount by which sample values are in error

Question 22

The correct formula for the standard error of the mean is

A)
B)
C)
D)

Question 23

From the formula for the standard error of the mean, it is apparent that variation among sample means will be decreased when

A) variation among scores in the population is less
B) sample size is larger
C) either (or both) of the above occurs
D) sampling is done with replacement

Question 24

The Central Limit Theorem

A) says that $\overline{X}$ approaches $\mu$ as sample size increases
B) says that S approaches $\sigma$ as sample size increases
C) says that both (a) and (b) will occur
D) refers to a matter other than those stated above