1

Chap 6 Confidence interval for the

population mean p341

Choose an SRS of size n from a

population having unknown mean

and known sta. dev. . A level C

confidence interval for is *x z

n

Here *z is the value of on the std.

normal curve with area C between *z

and *z . The interval is exact when the

population distribution is normal and

approximately correct for large n in

other cases.

In general CIs have the form

Estimate margin of error

In the above case

Margin of error(m) = *zn

2

Note: In the above formula for the CI

for the population mean, n is the std.

dev. of the sample mean ( X ) and it

can also be written as * . ( )x z Std Dev X

Here are three ways to reduce the

margin of error (and the width of

the CI)

- Use a lower level of confidence

(smaller C)

- Increase the sample size (n).

- Reduce .

3

Sample size for desired margin of

error p352

The CI for population mean will

have a specified margin of error m

when the sample size is 2

*zn m

Example:

A limnologist wishes to estimate the

mean phosphate content per unit

volume of lake water. It is known

from previous studies that the std.

dev. has a fairly stable value of 4mg.

How many water samples must the

limnologist analyze to be 90%

certain that the error of estimation

does not exceed 0.8 mg?

Ans:n = [1.645x4/0.8]^2=67.65 = 68

4

Ex:. You want to rent an

unfurnished one-bedroom apartment

for next semester. The mean

monthly rent for a random sample of

10 apartments advertised in the local

newspaper is $540. Assume that the

std. dev. is $80. Find a 95% CI for

the mean monthly rent for

unfurnished one-bedroom

apartments available for rent in this

community.

5

StatCrunch Commands

Stat > Z Statistics > One Sample >

with data

Ex.

Degree of Reading Power (DRP)

scores for 44 students. Assume that

the population standard deviation is

11.0.

95% CI for the population mean

score is given in the MINITAB

output below. DRP Scores

40 26 39 14 42 18 25 43 46 27

19 47 19 26 35 34 15 44 40 38

31 46 52 25 35 35 33 29 34 41

49 28 52 47 35 48 22 33 41 51

27 14 54 45

95% confidence interval results: μ : mean of Variable

Std. Dev. = 11

Variable n Sample Mean Std. Err. L. Limit U. Limit

var1 44 35.090908 1.6583124 31.840677 38.34114

6

Ex. A random sample of 85 students in

Chicago city high schools takes a course

designed to improve SAT scores. Based

on these students a 90% CI for the mean

improvement in SAT scores for all

Chicago high school students is

computed as (72.3, 91.4) points.

Which of the following statements are

true? a) 90% of the students in the sample

improved their scores by between 72.3

and 91.4 points.

b) 90% of the students in the population

improved their scores by between 72.3

and 91.4 points.

c) 95% CI will contain the value 72.3.

d) The margin of error of the 90% CI

above is 9.55.

e) 90% CI based on a sample of 340 (85

X 4) students will have margin of error

9.55/4.

7

f) If the sampling procedure were repeated

many times, with samples of 85 students

then approximately 95% of the resulting

confidence intervals would contain the

population mean.

8

Statistical tests for the population

mean ( known)

A significance test is a formal

procedure for comparing observed

data with a hypothesis whose truth

we want to assess. A hypothesis is

a statement about the parameters in

a population or model.

Null hypothesis p362

The statement being tested in a test

of significance is called the null

hypothesis. Usually the null

hypothesis is a statement of “no

effect” or “no difference”.

9

Each of the following situations requires

a significance test about a population

mean .

State the appropriate null hypothesis H0

and alternative hypothesis Ha in each

case.

(a) The mean area of the several thousand apartments in a new

development is advertised to be 1250

square feet. A tenant group thinks that

the apartments are smaller than

advertised. They hire an engineer to

measure a sample of apartments to test

their suspicion. (b) Larry's car averages 32 miles per

gallon on the highway. He now switches

to a new motor oil that is advertised as

increasing gas mileage. After driving

3000 highway miles with the new oil, he

wants to determine if his gas mileage

actually has increased.

10

(c) The diameter of a spindle in a small

motor is supposed to be 5 millimeters. If

the spindle is either too small or too

large, the motor will not perform

properly. The manufacturer measures the

diameter in a sample of motors to

determine whether the mean diameter

has moved away from the target. (a) H0: = 1250 ft

2; Ha: < 1250 ft

2 (b) H0: = 32 mpg; Ha: > 32 mpg (c) H0: = 5

mm; Ha: 5 mm.

Test Statistic p364

A test statistic measures

compatibility between the null

hypothesis and the data.

- We use it for the probability

calculation that we need for our test

of significance

- It is a random variable with a

distribution that we know.

11

Example

An air freight company wishes to

test whether or not the mean weight

of parcels shipped on a particular

route exceeds 10 pounds. A random

sample of 49 shipping orders was

examined and found to have average

weight of 11 pounds. Assume that

the std. dev. of the weights () is 2.8

pounds.

The null and alternative hypotheses

in this problem are:

: 10

0: 10

H

Ha

The test statistic for this problem is

the standardized version of X .

/XZ

n

12

11 10 2.5

2.8/ 49Z

Decision: ?

p-value p365

The probability, computed assuming

that H0 is true, that the test statistic

would take a value as extreme as or

more extreme than that actually

observed is called the p-value of the

test.

- The smaller the p-value the stronger

the evidence against H0 provided by

the data.

13

Significance level p367

The decisive value of the p is called

the significance level.

- It is denoted by .

Statistical significance p367

If the p-value is as small or smaller

than , we say that the data are

statistically significant at level .

14

Ex.

The Pfft Light Bulb Company claims

that the mean life of its 2 watt bulbs is

1300 hours. Suspecting that the claim

is too high, a consumer group

gathered a random sample of 64 bulbs

and tested each. He found the

average life to be 1295 hours. Test

the company's claim using = .01.

Assume = 20hours.

Ans: H0: mu = 1300 Ha: mu < 1300 z = (1295-1300)/(20/sqrt(64)) = -2 p-value = 0.0228 > 0.01 do not rej.

15

Ex

A standard intelligence examination

has been given for several years with

an average score of 80 and a standard

deviation of 7. If 25 students taught

with special emphasis on reading

skill, obtain a mean grade of 83 on the

examination, is there reason to believe

that the special emphasis changes the

result on the test? Use = .05.

H0: mu=80 Ha: mu not equal to 80 Z (83-80)/(7/5)= 2.14 p-value = 2x 0.0162 = 0.0324 < 0.05 rej H0

16

StatCrunch Commands

Stat > Z Statistics > One Sample >

with data

Ex. Degree of Reading Power (DRP)

scores for 44 students. The MINITAB

output for the test of : 32

0: 32

H

Ha

is given below.

Hypothesis test results: μ : mean of Variable

H0 : μ = 32

HA : μ > 32

Std. Dev. = 11

Variable n Sample Mean Std. Err. Z-Stat P-value

var1 44 35.090908 1.6583124 1.8638883 0.0312

17

Confidence Intervals and two-sided

tests p373

A level two-sided significance test

rejects a hypothesis 0 0:H exactly

when the value 0

falls outside the

1 confidence interval for .

Example

For the example above 95% CI

83 +/- 1.96 x (7/5) = (80.256, 85.744)

The value 80 is not in this interval and

so we reject H0: = 80 at the 5%

level of significance.

18

Use and abuse of Tests (6.3 p382)

- Test for mean we discussed works

for simple random samples.

- The sample must be from a normal

distribution or the sample size

must be large (CLT)

- Sample mean can be affected by

outliers and so the test

- Population standard dev must be

known

19

Strength of evidence vs decision-

making

-Best way to give result of a

significance test is to give P-value.

Shows strength of evidence against

null hyp. Enables reader to judge

whether evidence “strong enough”.

-must choose before looking at

data.

If rejecting null in favour of

alternative expensive, need strong

evidence to reject null. Use a smaller

.

= 0.05 is commonly used

20

Statistical and practical significance

(p383)

Example

When a null hypothesis (“no effect”

or no difference”) can be rejected at

some level there is good evidence

that an effect is present. But that

effect can be extremely small (may

not be even practically important)

p396

21

1) A study was carried out to investigate the

effectiveness of a treatment. 1000 subjects

participated in the study, with 500 being

randomly assigned to the "treatment group" and

the other 500 to the "control (or placebo) group".

A statistically significant difference was

reported between the responses of the two

groups (P <0 .005).

State whether the following statements are true

of false. a) There is a large difference between the effects

of the treatment and the placebo.

b) There is strong evidence that the treatment is

very effective.

c) There is strong evidence that there is some

difference in effect between the treatment and

the placebo.

d) There is little evidence that the treatment has

some effect.

e) The probability that the null hypothesis is true

is less than 0.005

22

Don’t ignore lack of significance

p384

Statistical inference is not valid for

all data sets p385

Need to use proper experimental

design and appropriate analysis, with

right kind of randomization.

In general, learn how data produced,

assess whether test/interval

meaningful.

Beware of searching for

significance p386

23

Decision Errors p393

The error made by rejecting the

null hypothesis when it is true is

called a type I error.

The probability of making a type I

error is denoted by .

The error made by accepting the

null hypothesis when it is false is

called a type II error.

The probability of making a type II

error is denoted by .

24

Significance and type I error

p396

The significance level of any

fixed level test is the probability of a

Type I error. That is is the

probability that the test will reject

the null hypothesis H0 when H0 is in

fact true.

Power and Type II error p496

The power of a fixed level test

against a particular alternative is

1 .

Note: Power = 1

= 0 0|1 ( . )P Acc H H is false

= 0 0|(Re )P jecting H H is false

25

Note: power increases as 0

increases.

Example We want to test

H0: = 300

Ha: < 300 at the 5% level of

significance.

The sample size is n = 6, and the

population is assumed to have a

normal distribution with = 3.

(a) Find the power of this test against

the alternative = 299.

(b) Find the power against the

alternative = 295.

MINITAB commands

26

Stats > Power and sample size > 1-

sample Z

Note: In MINITAB difference means

0 for 1-sided tests and

0 for

2-sided tests.

Power and Sample Size 1-Sample Z Test

Testing mean = null (versus < null)

Calculating power for mean = null +

difference

Alpha = 0.05 Assumed standard

deviation = 3

Sample

Difference Size Power

-1 6 0.203734

-5 6 0.992608

27

Power and Sample Size 1-Sample Z Test

Testing mean = null (versus < null)

Calculating power for mean = null +

difference

Alpha = 0.05 Assumed standard deviation = 3

Sample Target

Difference Size Power Actual Power

-1 78 0.9 0.903039