Section 8.4

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Section 8.4

Estimating Population Proportions






Example 8.19: Finding a Point Estimate for a Population Proportion

A graduate student wishes to know the proportion of American adults who speak two or more languages. He surveys 565 randomly selected American adults and finds that 226 speak two or more languages. Estimate the proportion of all American adults who speak two or more languages.






Example 8.19: Finding a Point Estimate for a Population Proportion (cont.)

SolutionUse as a point estimate for p.

We estimate the proportion of all American adults who speak two or more languages to be 40%.

226560 4

5

ˆ

.

xp

n

p̂Numerator: how many have the characteristic

Denominator: The sample size







Margin of Error of a Confidence Interval for a Population Proportion

When the sample taken is a simple random sample, the conditions for a binomial distribution are met, and the sample size is large enough to ensure that

the margin of error of a confidence interval for a population proportion is given by

ˆ ˆ5 and 1 5,np n p

2

ˆ ˆ1p pE z

n







Margin of Error of a Confidence Interval for a Population Proportion (cont.)

where is the critical value for the level of confidence, c = 1 − , such that the area under the standard normal distribution to the right of is equal to

p̂� is the sample proportion, and n is the sample size.

2z

2z

,2







Confidence Interval for a Population Proportion The confidence interval for a population proportion is given by

Where is the sample proportion, which is the point estimate for the population proportion, and E is the margin of error.

ˆ ˆ

orˆ ˆ,

p E p p E

p E p E

p̂






Example 8.20: Constructing a Confidence Interval for a Population Proportion

A survey of 345 randomly selected students at one university found that 301 students think that there is not enough parking on campus. Find the 90% confidence interval for the proportion of all students at this university who think that there is not enough parking on campus. SolutionStep 1: Find the point estimate.

First calculate the point estimate, ̂ .p30134

0.872465

4ˆ xp

n






Example 8.20: Constructing a Confidence Interval for a Population Proportion (cont.)

Step 2: Find the margin of error. Refer to the table of critical z-values to find Use the formula for the margin of error for proportions to calculate E.

2 0.10 2 0.05 1.645.z z z

2

0.872464 1 0.8

0.02954

724641.64

2

ˆ

5

ˆ

45

1

3

p pE z

n







Step 3: Subtract the margin of error from and add the margin of error to the point estimate. Subtracting the margin of error from the sample proportion and then adding the margin of error to the sample proportion gives us the following endpoints of the confidence interval.

ˆLower endpoint: 0.872464 0.029542

ˆUpper endpoin0.843

0t: 0.872464 0.029542

9

. 02

p E

p E







Thus, the 90% confidence interval for the population proportion ranges from 0.843 to 0.902. The confidence interval can be written mathematically using either inequality symbols or interval notation, as shown below.

or

0.

0.

843

8

0.

43

90

, 0

2

0.9 2

p







Note that these proportions are expressed as decimals, which is typical for confidence intervals. In this example, we can interpret the confidence interval (0.843, 0.902) to mean that, with 90% confidence, we estimate that between 84.3% and 90.2% of all students at this university think there is not enough parking on campus.






Example 8.21: Using a TI-83/84 Plus Calculator to Find a Confidence Interval for a Population Proportion

In a recent study, 820 American adults were surveyed and 61% said they would travel less this year for nonbusiness trips than last year. Find the 95% confidence interval for the true proportion of adults who said that they would travel less this year for nonbusiness trips. SolutionWe need three pieces of information to use the calculator to construct the confidence interval for the population proportion: n, x, and level of confidence. From the problem we know that n = 820 and c = 0.95.






Example 8.21: Using a TI-83/84 Plus Calculator to Find a Confidence Interval for a Population Proportion (cont.)

We only need to find x, or the number of adults who said they would travel less this year. You should note that the calculator wants a whole number for x and will return an error if you enter either a decimal or the percentage of the sample. In general, since is equivalent to the fraction of the sample that has the characteristic of interest, we can multiply

both sides of this equation by n to get a formula for x:

p̂

ˆ ,x

pn

ˆ .x p n







We were told that 61% of the adults surveyed said they would travel less, so we can find x by multiplying 0.61 by the sample size, 820, as follows.

Because we cannot enter a decimal, we can round this value to the nearest whole number. Approximately 500 adults have the characteristic of interest, so x = 500.

0.61 820

0

ˆ

5 0.2

x p n







To input these values into the calculator, press , select TESTS, and choose option A:1-PropZInt. Once entered, the calculator will give the results shown below.







Thus, the 95% confidence interval for the population proportion ranges from 0.576 to 0.643. The confidence interval can be written mathematically using either inequality symbols or interval notation, as follows.

We are 95% confident that the true proportion of American adults who say they will travel less this year for nonbusiness trips is between 57.6% and 64.3%.

or

0.

0.

576

5

0.

76

64

, 4

3

0.6 3

p






Minimum Sample Size for Estimating a Population Proportion


The minimum sample size required for estimating a population proportion at a given level of confidence with a particular margin of error is given by

2

21z

n p pE







Minimum Sample Size for Estimating a Population Proportion (cont.)

where p is the population proportion, is the critical value for the level of confidence,

such that the area under the standard normal

distribution to the right of is equal to and

E is the desired maximum margin of error.

2z

1 ,c 2z ,

2






Example 8.22: Finding the Minimum Sample Size Needed for a Confidence Interval for a Population Proportion

What is the minimum sample size needed for a 99% confidence interval for the population proportion if previous studies indicated p ≈ 0.54 and we desire no more than a 2% margin of error? SolutionTo find the minimum sample size, we need values for p, E, and One way to estimate p is from previous studies. Therefore, we will use p ≈ 0.54 as given. Since we’d like the margin of error to be at most 2%, we convert to a decimal to have E = 0.02.

2.z






Example 8.22: Finding the Minimum Sample Size Needed for a Confidence Interval for a Population Proportion (cont.)

Finally, we can look up the critical value in the table to find 2 0.01 2 0.005 2.575.z z z

22

2

1

2.5750.54 1 0.54

0.024117.6184

125118

zp p

En







Remembering to always round a minimum sample size up to the next larger whole number, we then need a sample size of 4118 or larger.When using a TI-83/84 Plus calculator, a more accurate value for can be calculated using the function, invNorm, under the DISTR menu. • Press and then . • Choose option 3:invNorm(. The function syntax for this option, when using the standard normal curve, is invNorm(area to the left of z).

2z







• The area to the left of a critical z-value, is equal to For this problem,

Thus, so we would enter invNorm(0.995).

2 ,z

1 2. 1 1 0.99 0.01.c 1 2 1 0.01 2 0.995,







The calculator returns a value of approximately 2.575829. Using this more accurate value for in the formula results in a minimum sample size of 4121. Notice that this value is close to the previous answer we calculated, but not the same.

2 ,z






Example 8.23: Finding the Minimum Sample Size, Point Estimate, and Confidence Interval for a Population Proportion

The state education commission wants to estimate the proportion of tenth-grade students in the state who read at or below the eighth-grade level. The commissioner of education believes that the proportion is around 0.21. a. How large a sample would be required to estimate the proportion of all tenth graders in the state who read at or below the eighth-grade level with an 85% level of confidence and an error of at most 0.03?






Example 8.23: Finding the Minimum Sample Size, Point Estimate, and Confidence Interval for a Population Proportion (cont.)

b. Suppose a sample of minimum size is chosen. Of these students, 84 read at or below the eighth-grade level. Using this information, compute the sample proportion of all tenth graders reading at

or below the eighth-grade level. c. Construct the 85% confidence interval for the

population proportion of tenth graders in the state reading at or below the eighth-grade level.







Solutiona. From the problem, we know that E = 0.03 and

p = 0.21. Using the table of critical z-values, we find that We can now use this

information to calculate the minimum sample size.

2 0.15 2 0.075 1.44.z z z

22

2

1

1.440.21 1 0.2 382.23361

0.03

zp p

En







Thus, the minimum sample size required is 383.(Note: If you use the invNorm function on a TI-83/84 Plus calculator to find the value of it will result in

Using this value in the above formula will result in a minimum sample size of 382, when rounded up to The next whole number.)

2 ,z

2 1.439531.z







b. To compute the sample proportion, we divide the number reading at or below the eighth grade ‑

level by the sample size.

843830. 1 21

ˆ

2 93

xn

p







(Note: If the value found using the critical value of z from a TI-83/84 Plus calculator, 382, is used for the minimum sample size in this calculation, the resulting sample proportion is ˆ 0.219895.)p







c. To construct the confidence interval, we need to calculate the margin of error. Substituting the appropriate values into the formula, we have the following.

2

ˆ ˆ1

0.219321 1 0.21932

0.03044

11.44

7383

pE

pz

n

Using 1-PropZInt is MUCH BETTER than using the primitive formula!Avoid the formula unless it’s needed to answer some question or to double-check 1-PropZInt.







Notice that the margin of error we calculated is very close to the value of E that we were willing to accept in part a. Since we used the minimum sample size calculated in part a., this is to be expected. Therefore, subtracting the margin of error from the sample proportion and then adding the margin of error to the sample proportion gives us the following endpoints of the confidence interval.







Thus, the 85% confidence interval for the population proportion ranges from 0.189 to 0.250.

ˆLower endpoint: 0.219321 0.030447

ˆUpper endpoin0.189

0t: 0.219321 0.030447

2

. 50

p E

p E







The confidence interval can be written mathematically using either inequality symbols or interval notation, as follows.

0.189 < p < 0.250 or

(0.189, 0.250) (Note: Using the values obtained from a TI-83/84 Plus calculator in parts a. and b. also results in a confidence interval of 0.189 < p < 0.250.)







In other words, the state education commission can be 85% confident that the proportion of tenth-grade students in the state who read at or below the eighth-grade level is between 18.9% and 25.0%.

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Section 8.4