Question

in a sample of 167 children selected randomly from one town, it is found that 37 of them suffer from asthma. at the 0.05 significance level, test the claim that the proportion of all children in the town who suffer from asthma is 11%.

Answer 1

The claim that the proportion of all children in the town who suffer from asthma is 11% is wrong/rejected. The proportion is higher than 11%

Explanation:

We test the hypothesis that the proportion of children who suffer from asthma is 11%

or initial assumption, p = 0.11

now, the null hyposthesis gives, H0 p = 0.11 (after the calculations)

otherwise, we reject the hypothesis if p does not equal 0.11

we calculate the point estimate of the population(lets call it q)

q = x/n where x is the people with asthma and n is the sample size,

in our case, x = 37 and n = 167,

so q = 0.22

now a is significance level, a = 0.05

now, since we are testing if p is not equal to 0.11, this is a two-sided test,

so we divide a by 2 to get, a/2 = 0.025

now we find the critical value associated with 0.025 by looking at a Z table, we find,

the values are +1.96,-1.96

now we find the Z value by,

`Z = (q - p)/√(p(1-p)/n)`

putting values, we find,

`Z = (0.22-0.11)/√(0.11(1-0.11)/167)`

for which we find Z = 4.54

now since Z - 4.54 is greater than the critical value i.e Z = 4.54 > 1.96,

we reject the null hypothesis H0 that p = 0.11 or that the proportion of children in the town who suffer from asthma is 11%.

(the proportion is greater than 11%)