Mathematics: A telephone service representative believes that the proportion of customers completely satisfied with their local telephone service is different between the Northeast and the Midwest. The representative s belief is based on the results of a survey. The survey included a random sample of 1200 northeastern residents and 1280 midwestern residents. 44% of the northeastern residents and 36% of the midwestern residents reported that they were completely satisfied with their local telephone service. Find

(1/2) / (1/8) =1/2 * 8/1 =8/2 reduces to 4...so she can make 4 batches of pancakes...

A telephone service representative believes that

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23.01.2022

a) The 80 % of confidence interval for the difference in two proportions.

(0.055 ,0.105)

b)The critical value that should be used in constructing intervals.

z₀.₂ = 1.28 ( from z-table) at 80 % of level of significance

Step-by-step explanation:

Step :1

Given data the survey included a random sample of 1200 northeastern residents and 1280 midwestern residents.

n₁ = 1200

n₂ = 1280

Given 44% of the northeastern residents reported that they were completely satisfied with their local telephone service.

The first proportion 'p₁' = 0.44

q₁ = 1- p₁ = 1- 0.44 = 0.56

Given 36% of the midwestern residents reported that they were completely satisfied with their local telephone service.

The second proportion 'p₂' = 0.36

q₂ = 1- p₂ = 1- 0.36 = 0.64

Step 2:-

The 80% confidence interval of p₁- p₂ are

(p₁- p₂ ± zα S.E (p₁- p₂)

where standard error of p₁- p₂

$S.E (p_{1}-p_{2} ) = \sqrt{\frac{p_{1}q_{1} }{n_{1} }+\frac{p_{2}q_{2} }{n_{2} } }$

substitute all values ,

$S.E (p_{1}-p_{2} ) = \sqrt{\frac{0.44X0.56 }{1200 }+\frac{0.36X0.64 }{1280 } }$

on calculation , we get

standard error of (p₁- p₂ ) = 0.0196

Step 3:-

a) The critical value that should be used in constructing intervals.

z₀.₂ = 1.28 ( from z-table) at 80 % of level of significance

Now the 80 % of confidence interval for the difference in two proportions.

(p₁- p₂ ± zα S.E (p₁- p₂)

((0.44-0.36) - 1.28 (0.0196) , 0.44-0.36 + 1.28 (0.0196))

(0.08 - 0.0250 ,0.08 + 0.0250)

(0.055 ,0.105)

Conclusion:-

the 80 % of confidence interval for the difference in two proportions.

(0.055 ,0.105)

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