Question

Are there more children diagnosed with Autism Spectrum Disorder (ASD) in states that have larger urban areas over states that are mostly rural? In the state of Pennsylvania, a fairly urban state, there are 245 eight year olds diagnosed with ASD out of 18,440 eight year olds evaluated. In the state of Utah a fairly rural state, there are 45 eight year olds diagnosed with ASD out of 2,123 eight year olds evaluated ("Autism and developmental," 2008). Is there enough evidence to show that the proportion of children diagnosed with ASD in Pennsylvania is more than the proportion in Utah? Test at the 1% level.

Answer 1

a)

p1cap = X1/N1 = 245/18440 = 0.0133
p1cap = X2/N2 = 45/2123 = 0.0212
pcap = (X1 + X2)/(N1 + N2) = (245+45)/(18440+2123) = 0.0141

Below are the null and alternative Hypothesis,
Null Hypothesis, H0: p1 = p2
Alternate Hypothesis, Ha: p1 > p2

Rejection Region
This is right tailed test, for α = 0.01
Critical value of z is 2.33.
Hence reject H0 if z > 2.33

Test statistic

z = (0.0133-0.0212)/sqrt(0.0141*(1-0.0141)*(1/18440 + 1/2123))
z = -2.9236

P-value Approach
P-value = 0.9983
As P-value >= 0.01, fail to reject null hypothesis.