Idea Transcript
Section 7.2 Confidence Intervals for Population Proportions
© 2012 Pearson Education, Inc. All rights reserved.
1 of 83
Section 7.2 Objectives • Find a point estimate for the population proportion • Construct a confidence interval for a population proportion • Determine the minimum sample size required when estimating a population proportion
© 2012 Pearson Education, Inc. All rights reserved.
2 of 83
Point Estimate for Population p Population Proportion • The probability of success in a single trial of a binomial experiment. • Denoted by p Point Estimate for p • The proportion of successes in a sample. • Denoted by x number of successes in sample pˆ= n= sample size read as “p hat” © 2012 Pearson Education, Inc. All rights reserved.
3 of 83
Point Estimate for Population p Estimate Population with Sample Parameter… Statistic Proportion: p pˆ Point Estimate for q, the population proportion of failures • Denoted by qˆ= 1 − pˆ • Read as “q hat”
© 2012 Pearson Education, Inc. All rights reserved.
4 of 83
Example: Point Estimate for p In a survey of 1000 U.S. adults, 662 said that it is acceptable to check personal e-mail while at work. Find a point estimate for the population proportion of U.S. adults who say it is acceptable to check personal e-mail while at work. (Adapted from Liberty Mutual) Solution: n = 1000 and x = 662
x 662 pˆ= = = 0.662 = 66.2% n 1000
© 2012 Pearson Education, Inc. All rights reserved.
5 of 83
Confidence Intervals for p A c-confidence interval for a population proportion p •
pˆ − E < p < pˆ + E
where E = Zα
2
pq ˆˆ n
•The probability that the confidence interval contains p is c.
© 2012 Pearson Education, Inc. All rights reserved.
6 of 83
Constructing Confidence Intervals for p In Words
In Symbols
1. Identify the sample statistics n and x. 2. Find the point estimate pˆ.
3. Verify that the sampling distribution of pˆ can be approximated by a normal distribution. 4. Find the critical value zc that corresponds to the given level of confidence c. © 2012 Pearson Education, Inc. All rights reserved.
pˆ =
x n
npˆ ≥ 5, nqˆ ≥ 5
Use the Standard Normal Table or technology. 7 of 83
Constructing Confidence Intervals for p In Words 5. Find the margin of error E. 6. Find the left and right endpoints and form the confidence interval.
© 2012 Pearson Education, Inc. All rights reserved.
In Symbols
E = Zα
2
pq ˆˆ n
Left endpoint: pˆ − E Right endpoint: pˆ + E Interval:
pˆ − E < p < pˆ + E
8 of 83
Example: Confidence Interval for p In a survey of 1000 U.S. adults, 662 said that it is acceptable to check personal e-mail while at work. Construct a 95% confidence interval for the population proportion of U.S. adults who say that it is acceptable to check personal e-mail while at work. Solution: Recall
pˆ = 0.662
qˆ =1 − pˆ =1 − 0.662 = 0.338
© 2012 Pearson Education, Inc. All rights reserved.
9 of 83
Solution: Confidence Interval for p • Verify the sampling distribution of pˆ can be approximated by the normal distribution
npˆ = 1000 ⋅ 0.662 =662 > 5 nqˆ = 1000 ⋅ 0.338 =338 > 5 • Margin of error:
pq (0.662) ⋅ (0.338) ˆ ˆ = ≈ 0.029 E Z= 1.96 α 2 n 1000
© 2012 Pearson Education, Inc. All rights reserved.
10 of 83
Solution: Confidence Interval for p • Confidence interval: Left Endpoint:
pˆ − E ≈ 0.662 − 0.029 = 0.633
Right Endpoint:
pˆ + E ≈ 0.662 + 0.029 = 0.691
0.633 < p < 0.691
© 2012 Pearson Education, Inc. All rights reserved.
11 of 83
Solution: Confidence Interval for p • 0.633 < p < 0.691 Point estimate
pˆ − E
pˆ
pˆ + E
With 95% confidence, you can say that the population proportion of U.S. adults who say that it is acceptable to check personal e-mail while at work is between 63.3% and 69.1%. © 2012 Pearson Education, Inc. All rights reserved.
12 of 83
Determining Sample Size � is known, • When 𝒑
2
Zα 2 pq ˆˆ n= 2 E
• Where 𝑍𝛼⁄2 is the value associated with the desired confidence level, and E is the desired margin of error. Round up to the next integer. 13
Sample Size for Estimating the Population Proportion � is unknown, we use • When 𝒑
2
Zα 2 0.25 n= 2 E 14
Example: Sample Size You are running a political campaign and wish to estimate, with 95% confidence, the population proportion of registered voters who will vote for your candidate. Your estimate must be accurate within 3% of the true population proportion. Find the minimum sample size needed if 1. no preliminary estimate is available. Solution: Because you do not have a preliminary estimate for pˆ , use pˆ = 0.5 and qˆ = 0.5. © 2012 Pearson Education, Inc. All rights reserved.
15 of 83
Solution: Sample Size • c = 0.95
zα/2 = 1.96 2
n
Zα 2 pq ˆˆ = 2 E
E = 0.03
[1.96]
2
(0.5)(0.5)
(0.03)
2
≈ 1067.11
Round up to the nearest whole number. With no preliminary estimate, the minimum sample size should be at least 1068 voters. © 2012 Pearson Education, Inc. All rights reserved.
16 of 83
Example: Sample Size You are running a political campaign and wish to estimate, with 95% confidence, the population proportion of registered voters who will vote for your candidate. Your estimate must be accurate within 3% of the true population proportion. Find the minimum sample size needed if 2. a preliminary estimate gives pˆ = 0.31. Solution: Use the preliminary estimate pˆ = 0.31
qˆ =1 − pˆ =1 − 0.31 = 0.69 © 2012 Pearson Education, Inc. All rights reserved.
17 of 83
Solution: Sample Size • c = 0.95
zα/2 = 1.96 2
n
Zα 2 pq ˆˆ = E2
E = 0.03
[1.96]
2
(0.31)(0.69)
(0.03)
2
≈ 913.02
Round up to the nearest whole number. With a preliminary estimate of pˆ = 0.31, the minimum sample size should be at least 914 voters. Need a larger sample size if no preliminary estimate is available. © 2012 Pearson Education, Inc. All rights reserved.
18 of 83
Section 7.3 Confidence Intervals for the Mean (𝝈 known)
© 2012 Pearson Education, Inc. All rights reserved.
19 of 83
Section 7.3 Objectives • Find a point estimate and a margin of error • Construct and interpret confidence intervals for the population mean • Determine the minimum sample size required when estimating μ
© 2012 Pearson Education, Inc. All rights reserved.
20 of 83
Point Estimate for Population μ Point Estimate • A single value estimate for a population parameter • Most unbiased point estimate of the population mean μ is the sample mean x Estimate Population with Sample Parameter… Statistic Mean: μ x
© 2012 Pearson Education, Inc. All rights reserved.
21 of 83
Example: Point Estimate for Population μ A social networking website allows its users to add friends, send messages, and update their personal profiles. The following represents a random sample of the number of friends for 40 users of the website. Find a point estimate of the population mean, µ. (Source: Facebook) 140 105 130 97 80 165 232 110 214 201 122 98 65 88 154 133 121 82 130 211 153 114 58 77 51 247 236 109 126 132 125 149 122 74 59 218 192 90 117 105 © 2012 Pearson Education, Inc. All rights reserved.
22 of 83
Solution: Point Estimate for Population μ The sample mean of the data is Σx 5232 = x = = 130.8 n 40
Your point estimate for the mean number of friends for all users of the website is 130.8 friends.
© 2012 Pearson Education, Inc. All rights reserved.
23 of 83
Interval Estimate Interval estimate • An interval, or range of values, used to estimate a population parameter. Left endpoint
146.5
x = 130.8
115.1
( 115
Right endpoint
Point estimate
120
125
130
135
) 140
145
150
Interval estimate
How confident do we want to be that the interval estimate contains the population mean μ? © 2012 Pearson Education, Inc. All rights reserved.
24 of 83
Level of Confidence Level of confidence (C-Level) • The probability that the interval estimate contains the population parameter. c is the area under the
C-level standard normal curve between the critical values.
𝛼/2
–𝑧
z=0
Critical values
zc
∝/2
z
Use the Standard Normal Table to find the corresponding z-scores.
The remaining area in the tails is 1 – c . © 2012 Pearson Education, Inc. All rights reserved.
25 of 83
Level of Confidence • If the level of confidence is 90%, this means that we are 90% confident that the interval contains the population mean μ. C-Level = 0.90 α/2= 0.05
α /2= 0.05 –zα/2−=zc – 1.645
z=0
Zzα/2 c = 1.645
z
The corresponding z-scores are ±1.645. © 2012 Pearson Education, Inc. All rights reserved.
26 of 83
Sampling Error Sampling error • The difference between the point estimate and the actual population parameter value. • For μ: the sampling error is the difference x – μ μ is generally unknown x varies from sample to sample
© 2012 Pearson Education, Inc. All rights reserved.
27 of 83
Margin of Error Margin of error • The greatest possible distance between the point estimate and the value of the parameter it is estimating for a given level of confidence, c. • Denoted by E.
E = zα /2
σ n
When n ≥ 30, the sample standard deviation, s, can be used for σ.
• Sometimes called the maximum error of estimate or error tolerance. © 2012 Pearson Education, Inc. All rights reserved.
28 of 83
Example: Finding the Margin of Error Use the social networking website data and a 95% confidence level to find the margin of error for the mean number of friends for all users of the website. Assume the sample standard deviation is about 53.0.
© 2012 Pearson Education, Inc. All rights reserved.
29 of 83
Solution: Finding the Margin of Error • First find the critical values 0.95 0.025
0.025 zc –zα/2 =−–1.96
z=0
zα/2zc = 1.96
z
95% of the area under the standard normal curve falls within 1.96 standard deviations of the mean. (You can approximate the distribution of the sample means with a normal curve by the Central Limit Theorem, because n = 40 ≥ 30.) © 2012 Pearson Education, Inc. All rights reserved.
30 of 83
Solution: Finding the Margin of Error
E = Zα 2
σ
≈ Zα 2
n 53.0 ≈ 1.96 ⋅ 40 ≈ 16.4
s n
You don’t know σ, but since n ≥ 30, you can use s in place of σ.
You are 95% confident that the margin of error for the population mean is about 16.4 friends. © 2012 Pearson Education, Inc. All rights reserved.
31 of 83
Confidence Intervals for the Population Mean A c-confidence interval for the population mean μ • x−E