#### Transcript Section 6-4

E LEMENTARY S

### TATISTICS

**Section 6-5 Estimating a Population Proportion**

M ARIO F . T RIOLA E IGHTH Chapter 6. Section 6-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman E DITION 1

**Assumptions**

**1. The sample is a simple random **

*sample. *

**2. The conditions for the binomial distribution are satisfied 3. The normal distribution can be used to approximate the distribution of sample are both satisfied.**

**ˆ**

**5 and nq ˆ**

**5 **

Chapter 6. Section 6-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 2

**Notation for Proportions**

*p =*

**population proportion**

*p*

**ˆ**

*= x n*

**sample proportion of **

*x*

**successes in a sample of size **

*n*

**(pronounced ‘p-hat’) ˆ**

*= *

**1 ˆ**

*p =*

**sample proportion of **

*x*

**failures in a sample size of **

*n*

Chapter 6. Section 6-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 3

**Definition**

**Point Estimate The sample proportion **

*p*

**ˆ**

**is the best point estimate of the population proportion **

*p*

**.**

**(In other books population proportion can be noted as **

**)**

Chapter 6. Section 6-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 4

**Margin of Error of the Estimate of ***p*

*p*

**ME =**

*z*

###

**ˆ ˆ**

*n*

Chapter 6. Section 6-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 5

*p*

*p*

**ˆ**

**Confidence Interval for Population Proportion where**

**ME =**

*z*

###

**ˆ ˆ**

*n*

Chapter 6. Section 6-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 6

**Confidence Interval for **

*p*

*p*

**ˆ**

**Population Proportion**

**E (**

*p*

*p*

**ˆ**

**- E, **

**ˆ**

*p*

*p*

**+ E)**

Chapter 6. Section 6-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 7

**Round-Off Rule for Confidence Interval Estimates of ***p* Round the confidence interval limits to three significant digits.

*p*Round the confidence interval limits to three significant digits.

Chapter 6. Section 6-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 8

### Do people lie about voting?

• In a survey of 1002 people, 701 people said they voted in a recent election. Voting records showed that 61% of eligible voters actually did vote. Using these results, find the following about the people who “said” they voted: • a) Find the point estimate • b) Find the 95% confidence interval estimate of the proportion • c) Are the survey results are consistent with the actual voter turnout of 61%?

Chapter 6. Section 6-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 9

### Do people lie about voting?

• In a survey of 1002 people, 701 people said they voted in a recent election. Voting records showed that 61% of eligible voters actually did vote. Using these results, find the following about the people who said they voted: • a) Find the point estimate

*n*

701 1002 0.6996

0.700

Chapter 6. Section 6-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 10

### Do people lie about voting?

• b) Find the 95% confidence interval estimate of the proportion

*E*

*z*

/ 2

*n*

1.96

(.70)(.30) .0284

1002 .671

*E p p*

.728

*E*

Chapter 6. Section 6-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 11

### Do people lie about voting?

• c) Are the survey results consistent with the actual voter turnout of 61%?

We are 95% confident that the true proportion of the people who said they vote is in the interval 67.1%

Because 61% does not fall

*inside*

the interval, we can conclude our survey results are

*not*

consistent with the actual voter turnout.

Chapter 6. Section 6-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 12

**Determining Sample Size**

**ME **

**=**

*z*

###

*n*

**(solve for **

*n*

**by algebra)**

*n*

**=**

(

*z*

)

**2 ME 2 ˆ**

*p q*

**ˆ**

Chapter 6. Section 6-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 13

**Sample Size for Estimating Proportion p**

*n*

**=**

(

*z*

)

**2 E 2 ˆ**

*p q*

**ˆ When no estimate of p is known:**

*n*

**=**

(

*z*

)

**2 0.25**

**E 2**

Chapter 6. Section 6-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 14

**Two formulas for proportion sample size**

*n*

**=**

(

*z*

)

**2 ME 2**

*n*

**=**

(

*z*

)

**2**

(

**0.25**

)

**ME 2**

Chapter 6. Section 6-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 15

**Example: We want to determine, with a margin of error of four percentage points, the current percentage of U.S. households using e-mail. Assuming that we want 90% confidence in our results, how many households must we survey? A 1997 study indicates 16.9% of U.S. households used e-mail. **

Chapter 6. Section 6-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 16

**Example: We want to determine, with a margin of error of four percentage points, the current percentage of U.S. households using e-mail. Assuming that we want 90% confidence in our results, how many households must we survey? A 1997 study indicates 16.9% of U.S. households used e-mail. n = [z**

**/2 ] 2 ˆ p q E 2**

Chapter 6. Section 6-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 17

**Example: We want to determine, with a margin of error of four percentage points, the current percentage of U.S. households using e-mail. Assuming that we want 90% confidence in our results, how many households must we survey? A 1997 study indicates 16.9% of U.S. households used e-mail. n = [z**

**/2 ] 2 ˆ p q E 2 = [1.645] 2 (0.169)(0.831) 0.04**

**2**

Chapter 6. Section 6-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 18

**Example: We want to determine, with a margin of error of four percentage points, the current percentage of U.S. households using e-mail. Assuming that we want 90% confidence in our results, how many households must we survey? A 1997 study indicates 16.9% of U.S. households used e-mail. n = [z**

**/2 ] 2 ˆ p q E 2 = [1.645] 2 (0.169)(0.831) 0.04**

**= 237.51965**

**2 = 238 households To be 90% confident that our sample percentage is within four percentage points of the true percentage for all households, we should randomly select and survey 238 households.**

Chapter 6. Section 6-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 19

**Example: We want to determine, with a margin of error of four percentage points, the current percentage of U.S. households using e-mail. Assuming that we want 90% confidence in our results, how many households must we survey? There is no prior information suggesting a possible value for the sample percentage.**

Chapter 6. Section 6-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 20

**Example: We want to determine, with a margin of error of four percentage points, the current percentage of U.S. households using e-mail. Assuming that we want 90% confidence in our results, how many households must we survey? There is no prior information suggesting a possible value for the sample percentage.**

**n = [z**

**/2 ] 2 (0.25) E 2**

Chapter 6. Section 6-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 21

**Example: We want to determine, with a margin of error of four percentage points, the current percentage of U.S. households using e-mail. Assuming that we want 90% confidence in our results, how many households must we survey? There is no prior information suggesting a possible value for the sample percentage.**

**n = [z**

**/2 ] 2 (0.25) E 2 = (1.645) 2 (0.25) 0.04**

**2 = 422.81641 = 423 households**

Chapter 6. Section 6-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 22

**n = [z**

**/2 ] 2 (0.25) E 2 = (1.645) 2 (0.25) 0.04**

**2 = 422.81641 = 423 households With no prior information, we need a larger sample to achieve the same results with 90% confidence and an error of no more than 4%.**

Chapter 6. Section 6-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 23

**Finding the Point Estimate and E from a Confidence Interval Point estimate of ˆ**

*p *

**= ˆ**

*p*

**: (upper confidence interval limit) + (lower confidence interval limit) 2 Margin of Error: E = (upper confidence interval limit) - (lower confidence interval limit) 2**

Chapter 6. Section 6-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 24

**Finding the Point Estimate and E from a Confidence Interval Given the confidence interval .214< p < .678**

**Find the point estimate of ˆ**

*p*

**: ˆ**

*p *

**= .678 + .214**

=

**.446**

**2 Margin of Error: **

*E *

**= .678 - .214**

=

**.232**

**2**

Chapter 6. Section 6-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 25