Chapter 8: Sampling Distributions

Question 1

Sampling Distribution

Answer 1

A probability distribution for all possible values of the statistic computed from a sample of size n.

Question 2

Sampling Distribution of the Sample Mean x¯

Answer 2

The probability distribution of all possible values of the random variable x¯ computed from a sample of size n from a population with mean μ and standard deviation σ.

Question 3

Steps for Determining the Sampling Distribution of the Sample Mean

Answer 3

Step 1: Obtain a simple random sample of size n.
Step 2: Compute the sample mean.
Step 3: Assuming that we are sampling from a finite population, repeat Steps 1 and 2 until all distinct simple random samples of size n have been obtained.

NOTE: Once a particular sample is obtained, it cannot be obtained a second time.

Question 4

The Shape of the Sampling Distribution of x¯¯ If X Is Normal

Answer 4

If a random variable X is normally distributed, the distribution of the sample mean, x¯, is normally distributed.

Question 5

Standard Error of the Mean

Answer 5

The standard deviation of the sampling distribution of x¯,σx¯
σx¯ = σ/√n.

Question 6

Describing the Distribution of the Sample Mean: Normal Population Using TI-84 Calculator

Answer 6

1. If the population is normally distributed, the distribution of the sample mean will be as well.
2. Determine if the sample size, n, is less than 10% of the population
3. determine mean of x-bar
4. Calculate standard deviation of x-bar
5. Open the calculator:
   - 2nd VARS (Distribution)
   - Option 2: normalcdf (
   - upper:
   - lower:
   - paste
   - enter
   - enter

Question 7

Describing the Distribution of the Sample Mean: Normal Population Using Microsoft Excel

Answer 7

1. Determine mean of x-bar
2. Calculate standard deviation of x-bar
3. For “less than” area:
   
   • type “NORM.DIST(x,mean,standard deviation,TRUE)”
   • press ENTER
4. For “greater than” area:
   
   • type “1-NORM.DIST(X,MEAN,st.dev.,TRUE)”
   • press ENTER
5. For “middle” area:
   
   • type “NORM.DIST(b,mean,st.dev.,TRUE) -
     “NORM.DIST(a,MEAN,ST.DEV.,TRUE), where a=smaller
     value and b=larger value.

Question 8

Central Limit Theorem

Answer 8

Regardless of the shape of the population, the sampling distribution of x¯ becomes approximately normal as the sample size n increases.

Question 9

A Rule of Thumb for Invoking the Central Limit Theorem

Answer 9

If the distribution of the population is unknown or not normal, then the distribution of the sample mean is approximately normal provided that the sample size is greater than or equal to 30.

Question 10

Shape, Center, and Spread of the Distribution of the Sample Mean for a Normal Population with Mean _ and Standard Deviation _

Answer 10

Shape: Regardless of the sample size n, the shape of the distribution of the sample mean is normal.

Center: The mean of the distribution is equal to the mean of the population.

Spread: The standard deviation of the distribution is equal to the standard deviation of the population divided by the square root of the sample size.

Question 11

Shape, Center, and Spread of the Distribution of the Sample Mean for a Population That Is Not Normal, with Mean _ and Standard Deviation _

Answer 11

Shape: As the sample size n increases, the shape of the distribution of the sample mean becomes approximately normal.

Center: The mean of the distribution is equal to the mean of the population.

Spread: The standard deviation of the distribution is equal to the standard deviation of the population divided by the square root of the sample size.

Question 12

Sample Proportion

Answer 12

The proportion of individuals in a sample who have a specified characteristic.

Given by p ̂ = x/n, where x is the number of individuals in the sample with the specified characteristic. The sample proportion, pˆ, is a statistic that estimates the population proportion, p.

NOTE: x can be thought of as the number of successes in n trials of a binomial experiment.

Question 13

Describe the Distribution of the Sample Mean: Normal Population

Answer 13

1. The random variable X is normally distributed, so the sampling distribution of x¯ will also be normally distributed.
2. Verify the sample size is less than 5% of the population size (the independence requirement).
3. The mean of the sampling distribution is μx¯=μ, and its standard deviation is σx¯ = √σ/ n.
4. To find the probability by hand, convert the sample mean x¯ to a z-score and then find the area under the standard normal curve to the right, left, or in-between of this z-score, or use technology to find the area.

Question 14

Rules for Describing the Distribution of the Sample Proportion p ̂ (p-hat).

Answer 14

For a simple random sample of size n with a population proportion p:

1. The shape of the sampling distribution of pˆ is approximately normal provided np (1−p) ≥ 10.
2. The mean of the sampling distribution of pˆ is μpˆ=p.
3. The standard deviation of the sampling distribution of pˆ is σpˆ = √p(1−p)/n.

NOTE: The sample size, n, can be no more than 5% of the population size, N. That is, n ≤ 0.05 N.