Lecture 7 - Confidence Intervals and Hypothesis Testing

Lecture 7 - Confidence Intervals and Hypothesis Testing

Statistics 102

Colin Rundel

February 6, 2013

From last time

CLT - Conditions

Certain conditions must be met for the CLT to apply:

1

Independence: Sampled observations must be independent.

This is difficult to verify, but is more likely if

random sampling/assignment is used, and

n < 10% of the population.

2

Sample size/skew: the population distribution must be nearly normal

or n > 30 and the population distribution is not extremely skewed.

This is also difficult to verify for the population, but we can check it

using the sample data, and assume that the sample mirrors the

population.

Statistics 102 (Colin Rundel)

Lec 7

February 6, 2013

2 / 26

From last time

Example - Review

To the right is a plot of a population distribution.

Match each of the following descriptions to one of

the three plots below.

Population

? = 10

��=7

(1) a single random sample of 100 observations from

this population

(2) a distribution of 100 sample means from random

samples with size 7

(3) a distribution of 100 sample means from random

0

10

20

30

40

50

samples with size 49

30

25

20

15

10

5

0

30

25

20

15

10

5

0

4

6

8

10

12

14

16

Plot A

Statistics 102 (Colin Rundel)

18

20

15

10

5

0

0

5

10

15

20

Plot B

Lec 7

25

30

35

8

9

10

11

12

Plot C

February 6, 2013

3 / 26

Confidence intervals

Why do we report confidence intervals?

Confidence intervals

A plausible range of values for the population parameter is called a

confidence interval.

Using only a point estimate to estimate a parameter is like fishing in a

murky lake with a spear, and using a confidence interval is like fishing

with a net.

We can throw a spear where we saw a

fish but we are more likely to miss. If we

toss a net in that area, we have a better

chance of catching the fish.

If we report a point estimate, we probably will not hit the exact

population parameter. If we report a range of plausible values �C a

confidence interval �C we have a good shot at capturing the parameter.

Statistics 102 (Colin Rundel)

Lec 7

February 6, 2013

4 / 26

Confidence intervals

Constructing a confidence interval

Example - Relationships

A sample of 50 Duke students were asked how many long term exclusive

relationships they have had. The sample yielded a mean of 3.2 and a

standard deviation of 1.74. Estimate the true average number of exclusive

relationships using this sample.

The approximate 95% confidence interval is defined as

point estimate �� 2 �� SE

x? = 3.2

s = 1.74

s

1.74

SE = �� = �� �� 0.25

n

50

x? �� 2 �� SE = 3.2 �� 2 �� 0.25

= (3.2 ? 0.5, 3.2 + 0.5)

= (3.15, 3.25)

We are 95% confident that Duke students on average have been in

between 3.15 and 3.25 exclusive relationships

Statistics 102 (Colin Rundel)

Lec 7

February 6, 2013

5 / 26

 ................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download