9. Sampling Distribution of the Mean

Question 1

Sampling Distribution of the Mean

Answer 1

Probability distribution of means for all possible random samples of a given size from some population.

Question 2

What does ̅X represent?

Answer 2

Sample Mean

Samples = normal alphabet

Question 3

What does µ represent?

Answer 3

Population mean

Populations = Greek Alphabet

Question 4

What does μ(petit ̅X) represent?

mu sub X-bar

Answer 4

Sampling distribution of the mean

Question 5

What does s represent?

Answer 5

Sample standard deviation

Question 6

What does σ represent?

Answer 6

Population standard deviation

Question 7

What does σ(petit ̅X) represent?

sigma sub X-bar

Answer 7

Standard of error of the mean, or simply the standard error.

Question 8

Without peeking, list the special symbols for the:

• mean of the population
• mean of the sampling distribution of the mean
• mean of the sample
• standard error of the mean
• standard deviation of the sample
• standard deviation of the population.

Answer 8

• µ
• mu sub X-bar
• ̅X
• sigma sub X-bar
• s
• σ

Question 9

Mean of the Sampling Distribution of the Mean (µ sub X-bar)

Answer 9

The mean of all samples means always equals the population mean.

Question 10

True or False?

The mean of all sample means, µ sub X-bar, always equals the value of a particular sample mean.

Answer 10

False. It always equals the value of the population mean.

Question 11

True or False?

The mean of all sample means, µ sub X-bar, equals 100 if, in fact, the population mean equals 100.

Answer 11

True.

Question 12

True or False?

The mean of all sample means, µ sub X-bar, usually equals the value of a particular sample mean.

Answer 12

False. Because of chance, most sample means tend to be either larger or smaller than the mean of all sample means.

Question 13

True or False?

The mean of all sample means, µ sub X-bar, is interchangeable with the population mean.

Answer 13

True.

Question 14

Standard Error of the Mean (σ sub X-bar)

Answer 14

A rough measure of the average amount by which sample means deviate from the mean of the sampling distribution or from the population mean.

σ sub X-bar = σ / √n

Question 15

True or False?

The standard error of the mean (σ sub X-bar) roughly measures the average amount by which sample means deviate from the population mean.

Answer 15

True.

Question 16

True or False?

The standard error of the mean (σ sub X-bar) measures varibility in a particular sample.

Answer 16

False. It measures variability among sample means.

Question 17

True or False?

The standard error of the mean (σ sub X-bar) increases in value with larger sample sizes.

Answer 17

False. It decreases in value with larger sample sizes.

Question 18

True or False?

The standard error of the mean (σ sub X-bar) equals 5, givent that σ = 40 and n = 64

Answer 18

True.

Question 19

Central Limit Theorem

Answer 19

Regardelss of the population shape, the shape of the sampling distribution of the mean approximates a normal curve if the sample size is sufficiently large.

Question 20

True or False?

The central limit theorem states that, with sufficiently large sample sizes, the shape of the population is normal.

Answer 20

False. The shape of the population remains the same regardless of sample size.

Question 21

True or False?

The central limit theorem states that, regardeless of sample size, the shape of the sampling distributions of the mean is normal.

Answer 21

False. It requires that the sample size be sufficiently large - usually between 25 and 100.

Question 22

True or False?

The central limit theorem ensures that the shape of the sampling distribution of the mean equals the shape of the population.

Answer 22

False. It ensures that the shape of the sampling distribution approximates a normal curve, regardless of the shape of the population (which remains intact).

Question 23

True or False?

The central limit theorem applies to the shape of the sampling distribution - not to the shape of the population and not to the shape of the sample.

Answer 23

True.