population sample parameter: proportion p count mean median statistic: proportion count mean median

Population Sample

Parameter:

Proportion p

Count

Mean Median

Statistic:

Proportion

Count

Mean

Median

x

p̂

Estimate population proportion, with a confidence interval, from data of a random sample.

Population

Proportion = p

Sample

p̂proportion

p

p̂ p̂p̂

Population

Proportion = p

Sample

p̂proportion

p

There is 95% chance that will fall inside the interval

p̂

n

ppp

12

n

pp 12

n

pp 12

ND

Population

Sample

p̂proportion

Proportion = p

p


n

ppp

12ˆ

p

n

pp 12

n

pp 12

Population

Sample

p̂proportion

Proportion = p

p


n

ppp

ˆ1ˆ2ˆ

p

n

pp ˆ1ˆ2

n

pp ˆ1ˆ2

Open the Fathom file ‘estimate1.ftm’

The proportion of “yes” in the population is given by the slider value p. (In this example, p =0.75)

Assume that the population proportion is an unknown, and we are going to estimate it by suggesting a 95% confidence interval based on the data of one random sample.

Size of this sample is n = 20.

Summary TableSample of data

20

0.85

0.079843597

1.0096872

0.69031281

S1 = countS2 = sp

S3 = sp sp

count

S4 = sp sp sp

count+

S5 = sp sp sp

count

The proportion of “yes” in this sample is

The estimated standard deviation of

is

Margin of error = 2(0.080) = 0.160

The 95% confidence interval is

(0.85 – 0.16, 0.85 + 0.16) or

(0.69, 1.01)

85.0ˆ p

p̂

080.0

20

85.0185.0ˆ1ˆ

n

pp

Based on the result of this sample, we are 95% confident that the true proportion p lies between 0.69 and 1.01.

Note that the interval will vary from sample to sample, but if we repeat the sampling process indefinitely with samples of the same size, we will expect 95% of these intervals to capture the true proportion.

To shorten the interval, we have to increase the sample size.

0 0.2 0.4 0.6 0.8 1

95% conf. Int.

Note that the interval will vary from sample to sample, but if we repeat the sampling process indefinitely with samples of the same size, we will expect 95% of these intervals to capture the true proportion.

0 0.2 0.4 0.6 0.8 1

Confidence Intervals from different samples

Use an experiment record sheet to record more confidence intervals from other samples of the same size.

Some intervals may not be able to capture the true proportion.

To estimate with a larger sample, double click on the ‘Sample of Data’ collection to open its inspector and adjust the sample size.


80

0.775

0.046687123

0.86837425

0.68162575

S1 = countS2 = sp

S3 = sp sp

count

S4 = sp sp sp

count+

S5 = sp sp sp

count

The proportion of “yes” in this sample is

and the sample size is n = 80.


is

Margin of error = 2(0.0467) = 0.0934


(0.775 – 0.0934, 0.775 + 0.0934) or

(0.682, 0.868)

775.0ˆ pp̂

0467.0

80

775.01775.0ˆ1ˆ

n

pp

0 0.2 0.4 0.6 0.8 1

95% conf. Int.

Sample size = 20

0 0.2 0.4 0.6 0.8 1

95% conf. Int.

Sample size = 80

We are now 95% confident that the true proportion lies between 0.682 and 0.868. The interval is shorter when the sample size is increased from 20 to 80.

Example: Halloween Practices and Beliefs

An organization conducted a poll about Halloween practices and beliefs in 1999. A sample of 1005 adult Americans were asked whether someone in their family would give out Halloween treats from the door of their home, and 69% answered ‘yes’.

Construct a 95% confidence interval for p, the proportion of all adult Americans who planned to give out Halloween treats from their home in 1999.

Adapted from Rossman et al. (2001, p.433)

Sample size = 1005

Sample proportion = 0.69

Estimated standard deviation of sample proportions =

0146.0

1005

31.069.0ˆ1ˆ

n

pp

Margin of error = 2(0.0146) = 0.0292

95% confidence interval is 0.69 0.0292

We are 95% confident that the population proportion lies between 0.6608 and 0.7192.

0 0.2 0.4 0.6 0.8 1

95% conf. Int.

Example: Personal Goal

According to a survey in a university, 132 out of 200 first-year students in a random sample have identified “being well-off financially” as an important personal goal.

Give a 95% confidence interval for the proportion of all first-year students at the university who would identify being well-off as an important personal goal.

Adapted from Moore & Mccabe (1999, p.597)

Sample size = 200

Sample proportion = 132/200 = 0.66

Estimated standard deviation of sample proportions =

0335.0

200

34.066.0ˆ1ˆ

n

pp

Margin of error = 2(0.0335) = 0.067

95% confidence interval is 0.66 0.067

We are 95% confident that the population proportion lies between 0.593 and 0.727.

0 0.2 0.4 0.6 0.8 1

95% conf. Int.

Estimate population mean, with a confidence interval, from data of a random sample.

Population

Mean =

Sample

xmeanxmean

xmeanxmean

Population

Mean =

Sample

xmean

n

2

n

2

s.d. =


x

n

2

ND

Population

Sample

xmean

Mean =

n

2

n

2

s.d. =


nx

2

Population

Sample

xmean

Mean =

n

s2

n

s2

s.d. = s


n

sx

2

Open the Fathom file ‘estimate2.ftm’

This summary table record the true mean and standard deviation of the population, where are supposed to be unknowns.

Assume that the population mean is an unknown, and we are going to estimate it by suggesting a 95% confidence interval based on the data of one random sample.

Size of this sample is n = 20.


20

29.85

20.228367

4.5232004

38.896401

20.803599

S1 = countS2 = smS3 = ssd

S4 = ssd

count

S5 = sm ssd

count+

S6 = sm ssd

count

The sample mean and standard deviation are


is

Margin of error = 2(4.52) = 9.046


(29.85 – 9.05, 29.85 + 9.05) or

(20.80, 38.90)

23.20 and 85.29 sxx

523.420

23.20

n

s

Based on the result of this sample, we are 95% confident that the true mean lies between 20.80 and 38.90.

Note that the interval will vary from sample to sample, but if we repeat the sampling process indefinitely with samples of the same size, we will expect 95% of these intervals to capture the true mean.

To shorten the interval, we have to increase the sample size.

0 10 20 30 40 50

95% conf. Int.

Example: Protein Intake

A nutritional study produced data on protein intake for women. In a sample of n = 264 women, the mean of protein intake is

grams and the standard deviation is s = 30.5 grams.

Estimate the population mean and give a 95% confidence interval .

6.59x

Adapted from Bennett et al. (2001, p.401)

Estimated standard deviation of the sample means =

9.1264

5.30

n

s

Margin of Error = 2(1.9) = 3.8 grams

95% confidence interval is 59.6 3.8 grams

We can say with 95% confidence that the interval ranging from 55.8 grams to 63.4 grams contains the population mean.

Example: Body Temperature

A study by University of Maryland researchers investigated the body temperatures of n = 106 subjects. The sample mean of the data set is and the standard deviation for the

sample is .

Estimate the population mean body temperature with a 95% confidence interval.

Fx 20.98Fs 62.0

Adapted from Bennett et al. (2001, p.403)

Estimated standard deviation of the sample means =

06.0106

62.0

n

s

Margin of Error = 2(0.06 F) = 0.12F

95% confidence interval is 98.20 F 0.12F

We can say with 95% confidence that the interval ranging from 98.08F to 98.32F contains the population mean.

Normal Distribution

m – s mm – 2s m – 3s m + s m + 2s m + 3s

68%

95%

99.7%

population sample parameter: proportion p count mean median statistic: proportion count mean median

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