l>Chapter 9 answersChapter 9. Inferential statistics: from samples to populations

1. What is the relationship between sampling variability and standard errors?

Standard errors are measures of sampling variability.

You are watching: To cut the standard error of the mean in half, the sample size must be doubled.

2. Assume is 2.40 and the sample size is 36. What will become if you change the sample size to:

a. 72

There are two ways to do this.

1.) Solve for s:

is 2.40 and the sample size is 36, and since is defined as and estimated as

*
, the standard deviation must be:

*

Now plug the standard deviation into the equation and get the new standard error:

*

2.) is defined as If you change the sample size by a factor of c, the new will be

*

But since

*
you can see that:
*

and the new will be

*
times the old

This is the "inverse square root" relation between sample size and . For this example, when you make the sample size twice as big, the will be

*
times as big, or
*


b. 9

The new sample size is one fourth as big, so:

*
and the new standard error will be twice as large as the original one:
*

c. 144

The new sample is four times as big, so:

*
and the new standard error is half as large as the original one:
*

3. Assume is 3.60 and your estimate for is 9.00. Assuming your sample size does not change, what will be if you could change to:

a. 12.0

Changing from 9.0 to 12.0 will increase the standard error of the mean by 12/9 = 1.33, which will give you 4.8 instead of 3.6.

b. 4.5

Changing from 9.0 to 4.5 will decrease the standard error of the mean by 4.5/9 = 0.5, which will give you 1.8 instead of 3.6.

c. 13.5

Changing from 9.0 to 13.5 will increase the standard error of the mean by 13.5/9 = 1.5, which will give you 5.4 instead of 3.6.

4. If the sample"s standard deviation tells you how good the sample"s mean is as a description of the typical person in the sample, the standard error of the mean tell you how good the sample"s mean is as a description of what? In other words, if the sample"s standard deviation tells you how far the sample"s mean is from the typical person in the sample, the standard error of the mean tells you how far the sample"s mean is likely to be from what?

How far from the population"s mean.

5. Calculate for the following eleven samples:

ns
a.366.01.0000
b.368.01.3333
c.3612.02.0000
d.496.00.8571
e.498.01.1428
f.7212.01.4142
g.986.00.6061
h.988.00.8081
i.9812.01.2122
j.14412.01.0000
l.1448.00.6667

6. Examine the answers you obtained for question 5.

a. What effect does doubling the sample size have on when s doesn"t change?

It is an inverse square relation. Multiplying the sample size by 2 divides the standard error by the square root of 2. The new will be:

*

b. What effect does quadrupling the sample size have on when s doesn"t change?

Multiplying the sample size by 4 divides the standard error by the square root of 4. The new will be

*
. It will be half as large as the original.

c. What effect does doubling s have on when the sample size doesn"t change?

The standard error of the mean is directly proportional to the standard deviation. Doubling s doubles the size of the standard error of the mean.

e. What effect does increasing s have on when the sample size doesn"t change?

Increasing s increases the size of the standard error of the mean by the same factor.

7. Overall, what is the relation between sample size and ?

Bigger samples produce smaller standard errors. The relation is an inverse square root relation: increasing the sample size by a factor of C decreases the standard error by a factor of one over the square root of C.

8. Calculate for the following six pairs of samples:

sample 1sample 2
nsns
a.455.50455.501.1595
b.605.50605.501.0042
c.608.50608.501.5519
d.455.50605.501.0846
e.4511.004511.002.3190
f.1805.501805.500.5798

9. Examine the answers for question 8.

a. What effect does increasing s have on when the sample size doesn"t change?

When s increases, increases.

b. What effect does increasing the sample size have on when s doesn"t change?

When n increases,decreases.

c. What effect does increasing the size of one sample have on when s and the other sample size don"t change?

When one sample size increases, decreases.

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d. What effect does doubling the sample size have on when s doesn"t change?

The new will become the old × 0.707107

e.What effect does quadrupling the sample size have on when s doesn"t change?

The new will become the old × 0.5

f. Overall, what is the relation between sample size and ?

Bigger samples smaller standard errors. The relation is an inverse square root relation: increasing the sample size by a factor of C decreases the standard error by a factor of one over the square root of C.

10. How is the shape of the sampling distribution related to your ability to make confidence estimates?