Chapter 8. The normal curve and samples: sampling distributions

1. If you have a population with of 144.0, what will be for samples of size 100?

Since is defined as and = 144, will be:

 

2. If you have a sample of 32 people and the variance (calculated as ) is 7.75, what is your best estimate for the standard error of the mean? (Be careful: Make it the best estimate possible with the data you have.)

is defined as which uses the population standard deviation in the numerator. Your best estimate of population standard deviation should be calculated with (n-1) in the denominator, so you have to adjust the estimate of variance you have. If you multiply it by 32 and divide the result by 31 you will have converted it to

.

So, your estimate for is:

which gives you 2.82843 for your best estimate of . You use this number to estimate the standard error of the mean:

3. If you are drawing random samples of 50 cases from a population that is normally distributed, has a population mean of 247 and a population standard deviation of 60.5, is the sampling distribution of sample means normally distributed?

Yes. Sampling distributions of sample means are always normally distributed.

 

4. If you are drawing random samples of 50 cases from a population that has a distribution shaped like the drawing below, population mean of 247 and population standard deviation of 60.5, is the sampling distribution of sample means normally distributed? Why or why not?

Yes. Sampling distributions of sample means are always normally distributed. It doesn't matter what the population distribution looks like.

 

5. It the mean of the sampling distribution is 193.482, the standard error of the mean is 100.0, and the sample size is 50, what is the mean of the population?

It will be the same as the mean of the sampling distribution: 193.482

 

6. All standard errors are standard deviations.

a. Standard deviations of what?

Standard deviations of sampling distributions.

 

b. How are they different from ordinary standard deviations of the data in a sample?

Ordinary standard deviations describe the dispersion of the cases in your sample. Standard errors describe the dispersion in sampling distributions. Each case in a sampling distribution is a sample, not an individual person.

 

c. Standard deviations measure the dispersion in your data. What dispersion do standard errors measure?

Standard errors describe the dispersion in sampling distributions - sampling variability - differences in sample statistics from sample to sample.

 

7. What is the probability that your sample mean is within one standard error of the population mean?

z z to mean smaller area larger area
: : : :
0.99 0.3389 0.1611 0.8389
1.00 0.3413 0.1587 0.8413
1.01 0.3438 0.1562 0.8438

The probability is about 2 × 0.3413, or 68.26%

 

8. What is the probability that your sample mean is within 1.85 standard errors of the population mean?

z z to mean smaller area larger area
: : : :
1.84 0.4671 0.0329 0.9671
1.85 0.4678 0.0322 0.9678
1.86 0.4686 0.0314 0.9686

The probability is 46.78% × 2 = 93.56%

 

9. What is the probability that your sample mean is more than 1.66 standard errors away from the population mean?

z z to mean smaller area larger area
: : : :
1.65 0.4505 0.0495 0.9505
1.66 0.4515 0.0485 0.9515
1.67 0.4525 0.0475 0.9525

The probability is 4.85% × 2 = 9.7%