BPS - 3rd Ed. Chapter 131 Confidence intervals: the basics.

BPS - 3rd Ed. Chapter 13 1

Chapter 13

Confidence intervals: the basics

BPS - 3rd Ed. Chapter 13 2

Two general types of statistical inference– Confidence Intervals (introduced this chapter)– Tests of Significance (introduced next chapter)

Statistical Inference

BPS - 3rd Ed. Chapter 13 3

1. SRS from population

2. Normal distribution X~N(, ) in the population

3. Although the value of is unknown, the value of the population standard deviation is known

Starting Conditions

BPS - 3rd Ed. Chapter 13 4

Case Study

NAEP Quantitative Scores(National Assessment of Educational Progress)

Rivera-Batiz, F. L. (1992). Quantitative literacy and the likelihood of employment among young adults. Journal of

Human Resources, 27, 313-328.

The NAEP survey includes a short test of quantitative skills, covering mainly basic arithmetic and the ability to apply it to realistic problems. Young people have a better chance of good jobs and wages if they are good with numbers.

BPS - 3rd Ed. Chapter 13 5

Case Study

Given:– Scores on the test range from 0 to 500– Higher scores indicate greater numerical ability– It is known NAEP scores have standard deviation = 60.

In a recent year, 840 men 21 to 25 years of age were in the NAEP sample– Their mean quantitative score was 272 (x-bar).– On the basis of this sample, estimate the mean score µ in

the population of 9.5 million young men in this age range

NAEP Quantitative Scores

BPS - 3rd Ed. Chapter 13 6

Case StudyNAEP Quantitative Scores

1. To estimate the unknown population mean , use the sample mean = 272.

2. The law of large numbers suggests that will be close to , but there will be some error in the estimate.

3. The sampling distribution of has a Normal distribution with unknown mean and standard deviation:

x

x

x

n

60

8402.1

BPS - 3rd Ed. Chapter 13 7

Case StudyNAEP Quantitative Scores

BPS - 3rd Ed. Chapter 13 8

Case Study

NAEP Quantitative Scores4. The 68-95-99.7 rule

indicates that and are within two standard deviations (4.2) of each other in about 95% of all samples.

x

BPS - 3rd Ed. Chapter 13 9

Case StudyNAEP Quantitative Scores

So, if we estimate that lies within 4.2 of , we’ll be right about 95% of the time.

x

4.2x is a 95% confidence interval for µ

BPS - 3rd Ed. Chapter 13 10

NAEP Illustration (cont.)

The confidence interval has the formestimate ± margin of error

estimate (x-bar in this case) is our guess for unknown µ

margin of error (± 4.2 in this case) shows accuracy of estimate

4.2x is a 95% confidence interval for µ

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Level of Confidence (C)

Probability that interval will capture the true parameter in repeated samples; the “success rate” for the method

You can choose any level of confidence, but the most common levels are:– 90%– 95% – 99%

e.g., If we use 95% confidence, we are saying “we got this interval by a method that gives correct results 95% of the time” (next slide)

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Fig 13.4

Twenty-five samples from the same population gave 25 95% confidence intervals

In the long run, 95% of samples give an interval that capture the true population mean µ

BPS - 3rd Ed. Chapter 13 13

Take an SRS of size n from a Normal population with unknown mean and known standard deviation . A “level C” confidence interval for is:

Confidence IntervalMean of a Normal Population

n

σzx

Confidence level C 90% 95% 99%

Critical value z* 1.645 1.960 2.576

BPS - 3rd Ed. Chapter 13 14

Confidence IntervalMean of a Normal Population

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Case StudyNAEP Quantitative Scores

Using the 68-95-99.7 rule gave an approximate 95% confidence interval. A more precise 95% confidence interval can be found using the appropriate value of z* (1.960) with the previous formula

267.884=4.116272=1)(1.960)(2. x276.116=4.116272=1)(1.960)(2. x

We are 95% confident that the average NAEP quantitative score for all adult males is between 267.884 and 276.116.

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The margin of error is:

The margin of error gets smaller, resulting in more accurate inference,– when n gets larger– when z* gets smaller (confidence level gets

smaller)– when gets smaller (less variation)

How Confidence Intervals Behave

margin of error = z

n

BPS - 3rd Ed. Chapter 13 17

Case Study

NAEP Quantitative Scores

90% Confidence Interval268.5455=45453272=1)(1.645)(2. .x275.4545=45453272=1)(1.645)(2. .x

The 90% CI is narrower than the 95% CI.

95% Confidence Interval

267.884=4.116272=1)(1.960)(2. x276.116=4.116272=1)(1.960)(2. x

BPS - 3rd Ed. Chapter 13 18

Choosing the Sample Size

The confidence interval for the mean of a Normal population will have a specified margin of error m when the sample size is:

2

m

σzn

BPS - 3rd Ed. Chapter 13 19

Case Study

NAEP Quantitative ScoresSuppose that we want to estimate the population mean NAEP scores using a 90% confidence interval, and we are instructed to do so such that the margin of error does not exceed 3 points.

What sample size will be required to enable us to create such an interval?

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Case Study

NAEP Quantitative Scores

Thus, we will need to sample at least 1082.41 men aged 21 to 25 years to ensure a margin of error not to exceed 3 points. Note that since we can’t sample a fraction of an individual and using 1082 men will yield a margin of error slightly more than 3 points, our sample size should be n = 1083 men.

1082.413

)(1.645)(6022

m

σzn