Question

Researchers would like to know whether the proportions of elementary school children who are obese differ in rural and urban area. An earlier study found that 50% of urban school children and 45% of rural school children are obese. The researchers select 153 urban school children and 191 rural school children. Suppose [^(p)]₁ and [^(p)]₂ denote sample proportions of urban and rural school children respectively who are obese.
Answer all the questions below (where appropriate) as a fraction not as a percentage.

What is the expected proportion of obese among urban school children, i.e. expected value of [^(p)]₁? [Answer to two decimal places.]

A: 0.34

B: 0.43

C: 0.50

D: 0.78

E: 0.97

Tries 0/3

What is the standard deviation of proportion of obese among urban school children, i.e. σ([^(p)]₁)? [Answer to four decimal places.]

A: 0.0404

B: 0.1299

C: 0.1494

D: 0.3599

E: 0.8455

Tries 0/3

What is the expected proportion of obese among rural school children, i.e. expected value of [^(p)]₂? [Answer to two decimal places.]

A: 0.44

B: 0.45

C: 0.49

D: 0.60

E: 0.98

Tries 0/3

What is the standard deviation of proportion of obese among rural school children, i.e. σ([^(p)]₂)? [Answer to four decimal places.]

A: 0.0360

B: 0.1933

C: 0.4943

D: 0.5395

E: 0.8700

Tries 0/3

What is the expected difference of proportions of obese between urban and rural school children, i.e. expected value of [^(p)]₁ − [^(p)]₂? [Answer to two decimal places.]

A: 0.02

B: 0.03

C: 0.05

D: 0.07

E: 0.11

Tries 0/3

What is the standard deviation of difference of proportions of obese between urban and rural school children, i.e. σ([^(p)]₁ − [^(p)]₂)? [Answer to four decimal places.]

A: 0.0002

B: 0.0044

C: 0.0541

D: 0.4994

E: 0.8715

Tries 0/3

What is the probability that the difference of proportions of obese between urban and rural school children will be larger than 0.10? i.e. find P([^(p)]₁ − [^(p)]₂ > 0.10). [Answer to four decimal places.]

A: 0.0451

B: 0.1778

C: 0.2705

D: 0.3058

E: 0.3746

Answer 1

Expected proportion of obese among urban school children, p̂1 = 0.50

Answer C.

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Standard deviation of proportion of obese among urban school children, σ(p̂1) = √(p̂1 *(1- p̂1 )/n)

= √(0.50 *(1- 0.50)/153) = 0.0404

Answer A.

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Expected proportion of obese among rural school children, p̂2 = 0.45

Answer B.

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What is the standard deviation of proportion of obese among rural school children, σ(p̂2) = √(p̂2 *(1- p̂2 )/n)

= √(0.45 *(1- 0.45)/191) = 0.0360

Answer A.

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Expected difference of proportions of obese between urban and rural school children, p̂1- p̂2 =

= 0.50 - 0.45 = 0.05

Answer C

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Standard deviation of difference of proportions of obese between urban and rural school children, σ(p̂1- p̂2) = √(0.0404² + 0.0360²) = 0.0541     

Answer C      

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Probability that the difference of proportions of obese between urban and rural school children will be larger than 0.10 =

P(Z > (p̂1- p̂2- 0.10)/σ(p̂1- p̂2)

= P(Z> (0.05 - 0.10)/0.0541 )

= P(Z> 0.9242)

= 1 - P(Z< 0.9242)

using excel function:

= 1 - Norm.s.dist(0.9242, 1)

= 1- 0.8222 =  0.1778

Answer B