1

STT 200 – LECTURE 1, SECTION 2,4RECITATION 13

(11/27/2012)

TA: Zhen (Alan) Zhang

zhangz19@stt.msu.eduOffice hour: (C500 WH) 1:45 – 2:45PM Tuesday

(office tel.: 432-3342)Help-room: (A102 WH) 11:20AM-12:30PM, Monday, Friday

Class meet on Tuesday: 3:00 – 3:50PM A122 WH, Section 02

12:40 – 1:30PM A322 WH, Section 04

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OVERVIEW

We will discuss following problems:

Chapter 19 “Confidence Intervals for Proportions”

(Page 504)

Nos. 7, 8, 14, 22, 27, 28

All recitation PowerPoint slides available at here

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Review

Confidence interval for proportion:

The half-width of the interval, is the margin of error:

will increase as the confidence level increases (or as decreases).

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Review (cont’d)

Margin of error

0.0 0.2 0.4 0.6 0.8 1.0

0.0

00

.02

0.0

40

.06

0.0

80

.10

Margin of error versus proportion

for fixed confidence level = 95%, sample size = 100proportion

ma

rgin

of e

rro

r

40 60 80 100 120 140

0.0

80

.10

0.1

20

.14

0.1

60

.18

Margin of error versus sample size

for fixed confidence level = 95%, proportion = 0.5sample size

ma

rgin

of e

rro

r

0.80 0.85 0.90 0.95

0.0

70

.08

0.0

90

.10

0.1

10

.12

0.1

3

Margin of error versus confidence level

for fixed sample size = 100, proportion = 0.5confidence level

ma

rgin

of e

rro

r

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Review (cont’d)

From , solve for sample size:

So given margin of error, confidence level, we can calculate required

sample size. When is unknown since we have not collected any

samples yet, we use a conservative guess under the “worst” scenario,

that is, when reaches its maximal (when ), it corresponds to the

largest required sample size.

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Chapter 19 (Page 504): #7:

Which statements are true?

a) For a given sample size, higher confidence means a smaller margin

of error.

b) For a specified confidence level, larger samples provides smaller

margins of error.

c) For a fixed margin of error, larger samples provide greater

confidence.

d) For a given confidence level, halving the margin of error requires a

sample twice as large.

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Chapter 19 (Page 504): #8:

Which statements are true?

a) For a given sample size, reducing the margin of error will mean

lower confidence.

b) For a certain confidence level, you can get a smaller margin of

error by selecting a bigger sample.

c) For a fixed margin of error, smaller samples will mean lower

confidence.

d) For a given confidence level, a sample 9 times as large will make a

margin of error one third as big.

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Chapter 19 (Page 505): #14:

11% of a random sample of 1003 adults approved of attempts to clone a human.

a) Find the margin of error if we want 95% confidence.

b) Explain what that margin of error means.

The pollsters are 95% confident that the true population of adults who approve

of attempts to clone humans is within 1.9% of the estimated 11%.

c) If we only need to be 90% confident, will the margin of error be larger or

smaller? Explain.

Smaller, since the critical value decreases.

d) Find that margin of error.

e) In general, if all other aspects of the situation remain the same, would

smaller samples produce smaller or larger margin of error?

Larger.

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Chapter 19 (Page 506): #22:

A national health organization warns that 30% of middle school students

have been drunk. A local health agency randomly surveys 110 of 1212

students. Only 21 of them report having been drunk.

a) What proportion of the sample reported having been drunk?

b) Does this mean that this city’s youth are not drinking as much as the

national data would indicate? Explain.

Maybe, since the estimation is much lower than 0.3, but this is just one estimate.

c) Create 95% confidence interval for the proportion of local students who

have been drunk.

95% confidence interval is 0.19090.0734 = (0.1175, 0.2643)

d) Is there any reason to believe that the national level of 30% is not true

of the middle school students in this city?

The 95% confidence interval does not cover 0.3

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Chapter 19 (Page 506): #27:

In a random survey of 226 college students, 20 reported being “only” children.

Estimate the proportion of students nationwide.

a) Check conditions for constructing a confidence interval.

The students’ birth orders are likely to be independent. The sample was random and

consisted of less than 10% of the population. There were 20 successes and 206 failures

(both greater than 10).

b) Construct 95% confidence interval.

c) Interpret your interval.

We are 95% confident that between 5.15% and 12.55% of all college students are “only”

children.

d) Explain what “95% confidence” means in this context.

If we were to select repeated samples like this we’d expect about 95% of the confidence

intervals we created to contain the true proportion of all college students who are “only”

children.

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Chapter 19 (Page 506): #28:

74% of 1644 randomly selected college freshmen returned to college the

next year. Estimate the national freshman-to-sophomore retention rate.

a) Verify that the conditions are met.

It’s a random sample; both 74% and 26% of 1644 are greater than 10.

b) Construct a 98% confidence interval.

The critical value is 2.326. The interval is

c) Interpret your interval.

We’re 98% confident that between 71.48% and 76.52% of colleges freshman

return to college their sophomore years.

d) Explain what “98% confidence” means in this context.

If we were to select repeated samples like this we’d expect about 98% of the

confidence intervals we created to contain the true proportion of all college

freshmen who return to be sophomores.

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Closing remark : The true proportion is fixed point (not random), but unknown. It is the

confidence interval that you constructed that’s random. So saying “ has 95% chance of falling between the 95% confidence interval” makes no sense unless you are a Bayesian statistician. On the contrary, we can only say “the 95% confidence interval has 95% chance of covering the true proportion ”, or routinely, we say “If we draw samples of this size many times and construct confidence interval in this way, we’d expect to see approximately 95% of them cover ”.

If I tell you the confidence interval is , will you be able to recover what is and what is the margin of error?

Yes you can. is always the middle point of this interval in this case, so

and the margin of error is half of the width, or |endpoint-middle point|

13Thank you.