Copyright (c) Bani Mallick1 Stat 651 Lecture 5. Copyright (c) Bani Mallick2 Topics in Lecture #5...

Copyright (c) Bani Mallick 1

Stat 651

Lecture 5


Topics in Lecture #5 Confidence intervals for a population

mean when the population standard deviation is known.

Properties of confidence intervals: what things make them longer and shorter.

Sample size calculation for a population mean when the population standard deviation is known : a simple illustration of a method.


Book Sections Covered in Lecture #5

Chapter 5.1

Chapter 5.2

Chapter 5.3


Lecture 4 Review: Pr(X < c) for Normal Populations

Compute the z-score

Look up value in Table 1

c-μz=

σ


Lecture 4 Review: Pr(X > c) for Normal Populations

Compute the z-score

Look up the value for z in Table 1

Subtract this value from 1.0

c-μz=

σ


Lecture 4 Review: Inference

The sample mean is a random variable

Its own “population” mean is

It’s standard deviation is

Note how the standard deviation of the sample mean becomes smaller as the sample size becomes larger

More data = more precision!!!!!

σ/ n


Lecture 4 Review: Central Limit Theorem

The sample mean is a random variable

Its own “population” mean is

It’s standard deviation is

In “large enough” samples, the sample mean is very nearly normally distributed, i.e., has a bell--shaped histogram

σ/ n


Confidence Interval for a Population Mean

A considerable part of basic statistics is to make inferences about the population mean

It is impossible to know the value of exactly.

This is a key factoid: why do I say this with such certainty?



A considerable part of basic statistics is to make inferences about the population mean

It is impossible to know the value of exactly.

Because (almost) every sample will give you a unique sample mean, and that sample mean will not equal the population mean.



What we can do is to construct an interval of possible values for the population mean .

The interval is determined by how much “confidence” we want in saying that the population mean is in the interval.

The interval is always of the formfactor confidence



The confidence factor is determined by how much confidence we want in concluding that the population mean is actually in the interval

Which interval has higher confidence of including the population mean?

-100 to -50 OR

-150 to 0

factor confidence


Confidence Interval for a Population Mean: Formal

Method The first method assumes that the

population standard deviation is known.

Suppose we want to be 95% confident that our interval includes the population mean , i.e., the probability is 95% that the population mean is in the interval.

Here is the interval:

n96.1 to

n96.1


WOMEN’S INTERVIEW SURVEY OF HEALTH (WISH)

computed the reported mean caloric intake at the start of the study, and the mean reported caloric intake at the end

My random variable X was the change (difference)

My hypothesis is that the population mean of X is < 0. In other words, I think women report less calories the more they are asked about their diet (Hawthorne Effect).


WISH: Change in Caloric Intake

271N =

Change in mean Energ

2000

1000

0

-1000

-2000

-3000

217239

208

247

Does it look like a big change?Note that the

scale of the box plotis -3000 to 2000


WISH

The sample size is n = 271

The sample mean change = -180

I am going to pretend that the population standard deviation is = 600.

n96.1 to

n96.1


WISH: Change in Reported Caloric Intake

n = 271, = 600, = -180

95% CI = -251 to -109

71 n

96.1

251- 71 - 180- n

96.1

109- 71 180- n

96.1


Review

= 600, n = 271, = -180

Then, with 95% probability, true population mean change is in the interval from -251 to -109

The chance is 95% that the population mean change is between 251 and 109 calories lower

Is there a Hawthorne effect?


Confidence Intervals

You can construct a confidence interval for the population mean with any level of confidence.

Generally, people report the 95% CI, but sometimes they report the 90% and 99% confidence intervals.

This is easy to do via a formula, and even easier to do via SPSS.


Confidence Interval for a Population Mean when is

Known Want 90%, 95% and 99% chance of

interval including .

90%

95%

99%

n645.1 to

n645.1

n96.1 to

n96.1

n58.2 to

n58.2


Confidence Intervals There is a general formula given on page 200

If you want a (1-)100% confidence interval for the population mean when the population s.d. is known, use the formula

The term z is the value in Table 1 that gives probability 1 - /2.

= 0.10, z = 1.645: = 0.05, z = 1.96, = 0.01, z = 2.58

nz to

nz 2/2/


WISH

The sample size is n = 271

The sample mean change = -180

I am going to pretend that the population standard deviation is = 600.

I want a 99% confidence interval: z = 2.58

n58.2 to

n58.2



n = 271, = 600, = -180

99% CI = -274 to -86

94 n

58.2

274- 94 - 180- n

58.2

86- 94 180- n

58.2



99% CI = -274 to -86

The chance is 99% that the population mean change in reported caloric intake is between 274 and 86 calories

The chance is less than 1% that there is no change in the population mean.



99% CI = -274 to -86

95% CI = -251 to -109

Note that the 99% CI is longer than the 95% CI.

This is clear(!): the more confidence you want, the longer the CI has to be.

Put another way, the less willing you are to be wrong, the more conservative your claims.


Effect of Sample Size

95% CI = -251 to -109 with n = 271

If n = 1000, the 95% CI would be from -217 to -143

Note how the CI gets shorter in length as the sample size gets larger.

This is a general fact: the larger the sample size the shorter the CI.


Effect of Population Standard Deviation

95% CI = -251 to -109 with = 600

If = 2000, the 95% CI would be from -418 to +58

Note how the CI gets longer in length as the population standard deviation gets larger.

This is a general fact: the larger the population standard deviation the longer the CI.


Using SPSS to Construct CI

SPSS actually assumes that the population standard deviation is unknown: we will consider this case later.

Its default is a 95% CI

You can easily change to any level of confidence

SPSS demo using Wish Data


Sample Size Determination

In general, this is a relatively complex issue, depending very heavily on the experiment.

I will show you a simple calculation in the special case that the population standard deviation is known.

Of course, is not known in practice, and more complex methods are required, but this will give you a feel for the process.



The usual answer to “what sample size should I take” is “what can you afford”.

Remember, more precision with larger sample sizes

Less precision with smaller sample sizes



The length of a confidence interval is

2 x confidence factor

Thus, our 95% CI for WISH was -251 to -109, so that the length was 142 calories

What if I wanted the length to be 100 calories?

Then the CI would have to be

factor confidence

50



The length of the CI is

If I want the length of a confidence interval to be

2 x E

then I have to set

Now I do some algebra

nz factor confidence 2/

nz2 2/

nz2E2 2/



I want the length of a confidence interval to be

2 x E

then the sample size I need is 2

2/ Ezn



Consider WISH, where = 600. Suppose I want the confidence interval length of 95% CI to be 2xE = 100

E = 50, z = 1.96

55350

60096.1

Ezn

22

2/



Consider WISH, where = 600. Suppose I want the confidence interval length of 95% CI to be 2xE = 60

E = 30, z = 1.96

1,53730

60096.1

Ezn

22

2/



95% confidence

Length = 100, E = 50, n = 553

Length = 60, E = 30, n = 1,557

General fact: the more precise you want to be (shorter CI), the larger the sample size you will need.



General fact: the larger the population standard deviation, the larger the sample size you will need to have a CI of length 2xE


Reactiver Oxygen Species (ROS) Data

Rats fed with Fish oil enhanced diets

Response is the change in ROS for an animal when the cells are exposed to butyrate


ROS Data

20N =

Change in Response

14

12

10

8

6

4

2

0

-2

3

10


ROS Data Sample mean = 3.21

Sample size is n = 20

Pretend = 3.33

Then = 0.74

95% interval for population mean change is

[3.21 - 0.74 * 1.96, 3.21 + 0.74 * 1.96] = [1.76, 4.66]:

Does butyrate increase ROS? How certain are we?

n


ROS Data = 3.33, n = 20

95% interval for population mean change is

[1.76, 4.66]

The length of the CI is 2xE = 2.90

What sample size would I need to make the length of the CI = 1.00? Here 2xE = 1.00, E = 0.50, and 170

50.0

33.396.1

Ezn

22

2/

Copyright (c) Bani Mallick1 Stat 651 Lecture 5. Copyright (c) Bani Mallick2 Topics in Lecture #5...

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