Chpt 7 - Sampling Distributions

Question 1

The population mean (μ) is a

Parameter/statistic

and is

not fixed/fixed

Answer 1

The population mean (μ) is a parameter and is a fixed number

Question 2

The sample mean (x̄) is a

Parameter/statistic

and is

not fixed/fixed

Answer 2

The sample mean (x̄) is a statistic and is not fixed

Question 3

Why is the sample mean (x̄) not a fixed number?

Answer 3

Because it’s values are means of the samples randomly selected from the population, so it changes form sample to sample

Question 4

Why is the sample mean a continuous random variable?

Answer 4

Because its value is NOT fixed and changes depending on the sample that is randomly selected.

Question 5

What type of variable is a sample mean?

Answer 5

Continuous random variable, because its value is not fixed and depends on the sample that is randomly selected

Question 6

Because the sample mean (X̄) is not a fixed number, what do we need to know to measure the error that one may make when using a sample mean (x̄) to estimate the population mean?

Answer 6

Which values it can take and how likely it takes a specified value

Long story short, we need to describe the distribution of the sample mean (X̄).

Question 7

What does the distribution of the sample mean (X̄) depend on?

Answer 7

The sample size of the samples selected

The distribution of the variable under consideration

Question 8

For any sample size, if the distribution of X is normal, what is the distribution of the sample mean (X̄)?

Answer 8

Normal

Question 9

For a larger sample size, if the distribution of X is not normal, what is the distribution of the sample mean (X̄)?

Answer 9

Approximately normal

Question 10

For a smaller sample size, if the distribution of X is not normal, what is the distribution of the sample mean (X̄)?

Answer 10

Not normal

Question 11

What is the distribution of the sample mean (X̄) called?

Answer 11

Sampling distribution of the sample mean

Question 12

What is the distribution of the random variable under consideration X called?

Answer 12

Parent distribution

Question 13

What is the central limit theorem (CLT)?

Answer 13

For a relatively large sample size n, the sample mean (X̄) is approximately normally distributed regardless of the parent distribution

Question 14

How large should n be to be large enough if the parent distributions is not too extremely skewed?

Answer 14

n>30

Question 15

Assume that the random variable X under consideration has a mean (μ) and standard deviation (σ). What is the mean of X̄ of samples with sample size n? What about the standard deviation?

Answer 15

μx̄ = μ

σx̄ = σ/√n

Question 16

If the parent distribution is normal with mean (μ) and standard deviation (σ), what is the shape, mean and standard deviation of the sampling distribution?

Answer 16

Normal shape

μx̄ = μ

σx̄ = σ/√n

Question 17

If the parent distribution is not normal with mean (μ) and standard deviation (σ), what is the shape, mean and standard deviation of the sampling distribution when the sample size is large?

Answer 17

Shape is approximately normal

μx̄ = μ

σx̄ = σ/√n

Question 18

If the parent distribution is not normal with mean (μ) and standard deviation (σ), what is the shape, mean and standard deviation of the sampling distribution when the sample size is small?

Answer 18

Shape cannot be determined

μx̄ = μ

σx̄ = σ/√n

Question 19

The standard deviation of X̄ is determined by the sample size n. The larger the sample size n, the ______ the standard deviation of X̄.

Answer 19

smaller

Question 20

A variable of a population has a mean of 40 and standard deviation of 2.

If the variable is normally distributed, what is the sampling distribution of the sample mean for samples of size 4?

Answer 20

Normal distribution

μx̄ = μ = 40

σx̄ = σ/√n = 2/√4 = 2/2 = 1

Question 21

A variable of a population has a mean of 40 and standard deviation of 2.

If the variables distribution is unknown, what is the sampling distribution of the sample mean for samples of size 4?

Answer 21

Distribution cannot be determined (sample size is too small)

μx̄ = μ = 40

σx̄ = σ/√n = 2/√4 = 2/2 = 1

Question 21

A variable of a population has a mean of 40 and standard deviation of 2.

If the variables distribution is unknown, what is the sampling distribution of the sample mean for samples of size 100?

Answer 21

Approximately normal

μx̄ = μ = 40

σx̄ = σ/√n = 2/√100 = 2/10 = 0.2

Question 22

What is the rule of thumb for deciding whether the sample size is relatively large such that the sample mean is approximately normally distributed?

Answer 22

n > 30

Question 23

Roughly speaking, what property of the distribution of the variable under consideration determines how large the sample size must be for a normal distribution to provide an adequate approximation to the distribution of the sample mean?

Answer 23

Skewness

a larger skew needs a larger sample size

Question 24

What is the chance that the sample mean and population mean are exactly the same?

Answer 24

0, a sampling error always exists

Question 25

What is a sampling error?

Answer 25

The difference between the sample mean and population mean

Question 26

Suppose the population mean of the parent distribution is unknown. A sample mean of a sample with n=2 and a sample mean of a sample with n=30 are obtained to estimate the population mean. Which one is more likely to be closer to the population mean, i.e., have smaller sampling error when used for estimating the population mean?

Answer 26

The larger sample size, so n=30

Question 27

How do we determine the probability that the sampling error made in estimating the population mean will be within a certain range, such as +/-1

Answer 27

1. Determine the sample distribution shape, mean, and standard deviation
2. Standardize the x values for the +/- range to find the z values
3. Use table 2 to convert the z values to the probability
4. Subtract - value from the + value to determine the probability that the sample mean falls within the +/- value

Question 28

Say we are attempting to determine the sampling error is within +/-1. How would you standardize the values for x?

Answer 28

Z+ = (X+1-μ)/σ

Z- = (X-1-μ)/σ

Determine the probabilities for both z values

P(Z+) - P(Z-) = P(μ of sample +/-1)