Answer:
a) CI = ( - 0,0087 ; 0,0355)
b) No CI contains 0 meaning that statistically the proportions could be equal
Step-by-step explanation:
Boys Sample
Size sample n₁ = 546
x₁ = 87
proportion p₁ = 87/546 p₁ = 0,159 p₁ = 15,9 %
Girls sample
Size sample n₂ = 508
x₂ = 74
proportion p₂ = 74 / 508 p₂ = 0,1456 p₂ = 14,56 %
Hypothesis test:
Null hypothesis H₀ p₁ = p₂
Alternative Hypothesis Hₐ p₁ ≠ p₂
Confidence Interval CI = 90 % significance level α = 10 %
α = 0,1 and as the alñternative hypothesis indicates is a two-tail test
then α/2 = 0,05
z(score) for 0,05 is from z table z(c) = 1,64
Confidence Interval 90 % is:
CI = [( p₁ - p₂) ± z(c) * √ p*q * ( 1/n₁ + 1 / n₂ )
p = (x₁ + x₂ ) / n₁ + n₂
p = ( 87 + 74 ) / 546 + 508
p = 0,1527 then q = 1 - 0,1527 q = 0,8473
CI = 0,0134 ± √ 0,1527*0,8473 ( 1/546 + 1 / 508 )
CI = 0,0134 ± 0,0221
CI = ( - 0,0087 ; 0,0355)
CI contains 0 meaning that difference between the groups could be 0
Therefore we can conclude that proportion on both groups are different. We can´t reject H₀
We need to calculate z(s)
z(s) = [ p₁ - p₂ ] / √ p*q * ( 1/n₁ + 1 / n₂ )
p = (x₁ + x₂ ) / n₁ + n₂
p = ( 87 + 74 ) / 546 + 508
p = 0,1527 then q = 1 - 0,1527 q = 0,8473
z(s) = [ 0,159 - 0,1456] / √ 0,1527*0,8473 ( 1/546 + 1 / 508 )
z(s) = 0,0134/ √0,1294 ( 0,0018 + 0,00196 )
z(s) = 0,0134/ √0,1294 ( 0,0038)
z(s) = 0,0134/ 0,0221
z(s) = 0,61
p-value for z(s) = 0,61 p-value = 0,7291
p-value > 0,05
As p-value is bigger than 0,05 we have to accept H₀. We don´t have enough evidence to claim any difference between the two groups
