Confidence Intervals

Elementary Statistics

Larson Farber

Chapter

6

2

Estimating Parameters

Hypothesis Testing

Inferential Statistics-the branch of statistics that uses sample statistics to make inferences about population parameters.

Inferential Statistics

Applications of Inferential Statistics

3

Point Estimate

DEFINITION:A point estimate is a single value estimate for a population parameter. The best point estimate of the population mean is the sample mean .x

4

Example: Point Estimate

A random sample of airfare prices (in dollars) for a one-way ticket from Atlanta to Chicago. Find a point estimate for the population mean, .

99 102 105 105 104 95 100 114 108 103 94 105 101 109 103 98 96 98 104 87 101 106 103 90 107 98 101 107 105 94 111 104 87 117 101

The sample mean is77.101

35

3562

n

xx

The point estimate for the price of all one way tickets from Atlanta to Chicago is $101.77.

5

Interval Estimates

An interval estimate is an interval, or range of values used to estimate a population parameter

•101.77Point estimate

( )•101.77

The level of confidence, c is the probability that the interval estimate contains the population parameter

6

0 z

Sampling distribution x

For c = 0.95

0.950.0250.025

95% of all sample means will have standard scores between z = -1.96 and z = 1. 96

Distribution of Sample Means

When the sample size is at least 30, the sampling distribution for is normalx

-1.96 1.96

7

Maximum Error of Estimate

DEFINITION: Given a level of confidence, c, the maximum error of estimate E is the greatest possible distance between the point estimate and the value of the parameter it is estimating.

When n 30, the sample standard deviation, s can be used for .n

zzE cxc

Find E, the maximum error of estimate for the one-way plane fare from Atlanta to Chicago for a 95% level of confidence given s = 6.69

Using zc=1.96, s = 6.69 and n = 35,

22.235

69.696.1

nzE c

You are 95% confident that the maximum error of estimate is $2.22

8

Definition: A c-confidence interval for the population mean is

ExEx

Confidence Intervals for

Find the 95% confidence interval for the one-way plane fare from Atlanta to Chicago.

You found = 101.77 and E = 2.22x

•101.77( )

Left endpoint

55.9922.277.101 Ex

99.55

Right endpoint99.10322.277.101 Ex

103.99

With 95% confidence, you can say the mean one-way fare from Atlanta to Chicago is between $99.55 and $103.99

99.55 103.99

9

Sample SizeGiven a c-confidence level and an maximum error of estimate, E, the minimum sample size n, needed to estimate , the population mean is 2

E

zn c

You want to estimate the mean one-way fare from Atlanta to Chicago. How many fares must be included in your sample if you want to be 95% confident that the sample mean is within $2 of the population mean?

98.422

69.696.122

E

zn c

You should include at least 43 fares in your sample. Since you already have 35, you need 8 more.

10

0t

n =13d.f.=12c=90%

.90

The t-distribution

-1.782 1.782

The critical value for t is 1.782. 90% of the sample means with n = 12 will lie between t = -1.782 and t = 1.782

.05 .05

Sampling distribution x

If the distribution of a random variable x is normal and n < 30, then the sampling distribution of is a t-distribution with n-1 degrees of freedom.

x

11

Confidence Interval-Small Sample

In a random sample of 13 American adults, the mean waste recycled per person per day was 4.3 pounds and the standard deviation was 0.3 pound. Assume the variable is normally distributed and construct a 90% confidence interval for .

1. The point estimate is x = 4.3 pounds

2. The maximum error of estimate is 148.0

13

3.0782.1

n

stE c

Maximum error of estimate

n

stE c

12

Confidence Interval-Small Sample

1. The point estimate is x=4.3 pounds

2. The maximum error of estimate is

148.013

3.0782.1

n

stE c

4.15 < < 4.45

•4.3(

Left endpoint

152.4148.03.4 Ex

4.152)

Right endpoint

448.4148.03.4 Ex

4.448

With 90% confidence, you can say the mean waste recycled per person per day is between 4.15 and 4.45 pounds.

13

Confidence Intervals for Population Proportions

pq ˆ1ˆ q̂ is the point estimate for the proportion of failures where

If np 5 and nq 5 , the sampling distribution for is

normal.p̂

The point estimate for p, the population proportion of successes is given by the proportion of successes in a sample

nˆ

xp (Read as p-hat)

14

Confidence Intervals for Population Proportions

The maximum error of estimate, E for a c-confidence interval is:

n

qpzE c

ˆˆ

A c-confidence interval for the population proportion, p is

EppEp ˆˆ

15

Confidence Interval for p

In a study of 1907 fatal traffic accidents, 449 were alcohol related. Construct a 99% confidence interval for the proportion of fatal traffic accidents that are alcohol related.

1. The point estimate for p is 235.01907

449

nˆ

xp

765.0235.01ˆ q

2. 1907(.235) 5 and 1907(.765) 5, so the sampling distribution is normal.

3. 025.01907

)765)(.235(.575.2

ˆˆ

n

qpzE c

16

Confidence Interval for p In a study of 1907 fatal traffic accidents, 449 were alcohol related. Construct a 99% confidence interval for the proportion of fatal traffic accidents that are alcohol related.

0.21 < p < 0.26

(

Left endpoint

21.0025.0235.0ˆ Ep

.21•.235

)

26.0025.0235.0ˆ Ep

Right endpoint

.26

With 99% confidence, you can say the proportion of fatal accidents that are alcohol related is between 21% and 26%.

17

If you have a preliminary estimate for p and q the minimum sample size given a c-confidence interval and a maximum error of estimate needed to estimate p is:

Minimum Sample Size

E

zqpn cˆˆ

If you do not have a preliminary estimate, use 0.5 for both ˆ ˆandp q

18

You wish to estimate the proportion of fatal accidents that are alcohol related at a 99% level of confidence. Find the minimum sample size needed to be be accurate to within 2% of the population proportion.

You will need at least 4415 for your sample.

Example: Minimum Sample Size

E

zqpn cˆˆ =

With no preliminary estimate use 0.5 for ˆ ˆandp q

14.441402.0

575.2)5.0)(5.0(

2

19

You wish to estimate the proportion of fatal accidents that are alcohol related at a 99% level of confidence. Find the minimum sample size needed to be be accurate to within 2% of the population proportion. Use a preliminary estimate of p = 0.235

05.298002.0

575.2)765.0)(235.0(

2

n

With a preliminary sample you need at least n =2981for your sample.

Example: Minimum Sample Size

E

zqpn cˆˆ

20

0 10 20 30 40

The Chi-Square DistributionThe point estimate for 2 is s2 and the point estimate for is s. If the sample size is n, use a chi-square 2 distribution with n-1 d.f. to form a c-confidence interval.

Find R2 the right- tail critical value and L

2 the left-tail critical value for c = 95% and n = 17.

Area to the right of R2 is (1- 0.95)/2 = 0.025 and area to the right of L

2 is (1+ 0.95)/2 = 0.975

R2

=28.845L2

=6.908

6.908 28.845

.95

When the sample size is 17, there are 16 d.f.

21

2

22

2

2 )1()1(

LR

snsn

Confidence Intervals for

A c-confidence interval for a population variance is:

To estimate the standard deviation take the square root of each endpoint.

908.6

150)117(

845.28

150)117( 22

2

Find the square root of each endpoint

212480.50 < < 52113.49

$117.72 < < $228.28

2 and

You randomly select and record the prices of 17 CD players. The sample standard deviation is $150. Construct a 95% confidence interval for

2 and

You can say with 95% confidence that is between 12480.50 and 52113.49 and between $117.72 and $228.28.

2