Putting it Together:Putting it Together:The Four Step ProcessThe Four Step Process

8.2b

h.w: pg 497: 39, 43, 47

Target Goal: I can carry out the steps for constructing a confidence interval.I can determine the sample size required to obtain a level C confidence interval.

Review: Finding a Critical Value

Use Table A to find the critical value z* for an 80% confidence interval. Assume that the Normal condition is met.

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Since we want to capture the central 80% of the standard Normal distribution, we leave out 20%, or 10% in each tail. Search Table A to find the point z* with area 0.1 to its left.

So, the critical value z* for an 80% confidence interval is z* = 1.28.Try invnorm(990)

The closest entry is z = – 1.28.

z .07 .08 .09

– 1.3 .0853 .0838 .0823

– 1.2 .1020 .1003 .0985

– 1.1 .1210 .1190 .1170

The Four-Step Process

We can use the familiar four-step process whenever a problem asks us to construct and interpret a confidence interval.

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State: What parameter do you want to estimate, and at what confidence level?

Plan: Identify the appropriate inference method. Check conditions.

Do: If the conditions are met, perform calculations.

Conclude: Interpret your interval in the context of the problem.

State: What parameter do you want to estimate, and at what confidence level?

Plan: Identify the appropriate inference method. Check conditions.

Do: If the conditions are met, perform calculations.

Conclude: Interpret your interval in the context of the problem.

Confidence Intervals: A Four-Step Process

Ex. Binge Drinking in CollegeEx. Binge Drinking in College

• In a representative of 140 colleges and 17592 students, 7741 students identify themselves as binge drinkers.

• Considering this SRS, construct a 95% confidence interval for the proportion of students who identify themselves as binge drinkers.

Step 1Step 1: Identify the population of interest and the : Identify the population of interest and the parameter you want to draw a conclusion about. parameter you want to draw a conclusion about.

• State: We want to estimate the actual proportion of all college students who identify themselves as binge drinkers at a 95% confidence level.

7741ˆ 0.44

17,592p

Step 2Step 2: Choose the appropriate inference : Choose the appropriate inference procedure. Verify the conditions for using the procedure. Verify the conditions for using the

selected procedure.selected procedure.

Plan: We will use a one-sample z interval for p if the conditions are met.

• Random: SRS? Yes given. • Independent: Total population > 10 n:

10(17592) = there are more than 175,920 college students in the country ( to use sample σ) so yes

• Normal:17592(.44) = 7740 ≥ 1017592(.56) = 9851 ≥ 10

Yes, we can use normal approximation.

ˆ ˆ10 and 10 np nq

Step 3: DO -Step 3: DO - If the conditions are met, If the conditions are met, perform calculations.perform calculations.

Diagram:

invnorm(1-.025)

• z* = (table A or calc.)

Ex. Estimating Risky BehaviorEx. Estimating Risky Behavior

•

ˆ ˆˆ *

pqp z

n (0.44)(0.56)

0.44 1.9617592

0.44 0.0073 (0.4327,0.4473)CI

• Step 4: Conclude

We are 95% confident that the actual percent of all college students who identify themselves as binge drinkers lies between 43% and 45%.

Ex. Ex. Is that Coin Fair? Is that Coin Fair?

• The French naturalist Count Buffon tossed a coin 4040 times and counted 2048 heads. The sample proportion of heads is

• = 0.5069 p̂

Ex. Ex. Confidence Interval for pConfidence Interval for p

• Calculate the 95% C.I. for the probability p that Buffon’s coin gives a head.

• (Do and Conclude only)

•

ˆ ˆ(1 )ˆ *

p pp z

n

(0.5069)(0.4931)0.5069

40400.5069

1.

0.015

9

4

60

• = (.4915, 0.5223)• We are 95% confident that the probability of a

tossing ahead is between 0.4915 and 0.5223.

• Now try: STAT: TESTS:1-Prop Z Int

Choosing the Sample SizeIn planning a study, we may want to choose a sample size that allows us to

estimate a population proportion within a given margin of error.

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The margin of error (ME) in the confidence interval for p is

ˆ ˆ(1 )*

p pME z

n

z* is the standard Normal critical value for the level of confidence we want.

To determine the sample size n that will yield a level C confidence interval for a population proportion p with a maximum margin of error ME, solve the following inequality for n:

To determine the sample size n that will yield a level C confidence interval for a population proportion p with a maximum margin of error ME, solve the following inequality for n:

Sample Size for Desired Margin of Error

z *ˆ p (1 ˆ p )

nME

where ˆ p is a guessed value for the sample proportion. The margin of errorwill always be less than or equal to ME if you take the guess ̂ p to be 0.5.

ˆBecause the margin of error involves the sample proportion , we have to

when choosing . There guess the latter valu are two ways to do t s:e hi

p

n

ˆ• Use a guess for based on past experience or a pilot studypˆ• Use as the guess. ˆ 0.5 ME is larg when e 5s .t 0p p

• Example: Customer SatisfactionRead the example on page 493.

Determine the sample size needed to estimate p within 0.03 with 95% confidence.

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The critical value for 95% confidence is z* = 1.96.

Since the company president wants a margin of error of no more than 0.03, we need to solve the equation

ˆ ˆ(1 )1.96 0.03

p p

n

1.96ˆ ˆ(1 )

0.03p p n

21.96

ˆ ˆ(1 )0.03

p p n

21.96

(0.5)(1 0.5)0.03

n

1067.111 n

Multiply both sides by square root n and divide

both sides by 0.03.

Multiply both sides by square root n and divide

both sides by 0.03.

Square both sides.Square both sides.

Substitute 0.5 for the sample proportion to find the largest ME

possible.

Substitute 0.5 for the sample proportion to find the largest ME

possible.

We round up to 1068 respondents to ensure the

margin of error is no more than 0.03 at 95%

confidence.

We round up to 1068 respondents to ensure the

margin of error is no more than 0.03 at 95%

confidence.

ˆ ˆ(1 )*

p pz m

n

• Read pg. 490 - 494