Question

In each problem show all steps of the hypothesis test or confidence interval. If some of the assumptions are not met, note that the results of the test or interval may not be correct and then continue the process of the hypothesis test or confidence interaval.

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 ASDout of 18,440 eight year olds evaluated. In the state of Utah, a fairly rural state, there are 45 eight year olds diagnosed with ASDout of 2,123 eight year olds evaluated ("Autism and developmental," 2008). Is there enough evidence to show that the proportion of children diagnosed with ASDin Pennsylvania is more than the proportion in Utah? Test at the 1% level

Answer 1

Research Question:

Is there enough evidence to show that the proportion of children diagnosed with ASD in Pennsylvania is more than the proportion in Utah?

Let p₁ = Proportion of children diagnosed with ASD in Pennsylvania

p₂ = Proportion of children diagnosed with ASD in Utah

Null hypothesis:

H0: There is no significant difference between the proportion of children diagnosed with ASD between Pennsylvania and Utah, p₁ = p₂

Alternative hypothesis:

H1: The proportion of children diagnosed with ASD in Pennsylvania is more than the porportion of children diagnosed with ASD in Utah, p₁ > p₂

Level of significance:

Critical Region:

Since, the alternative hypothesis is testing on the one-side(researcher is interseted if proportion in one group is more than the other) the hypothesis is tested at one tail. Therefore, form the standard normal table for 1% level of significance the Z-value is given as 2.33. Therefore, the null hypothesis will be rejected if the Z-test statistic is more than 2.33

Test statistic:

Where

p₁ = Proportion of children diagnosed with ASD in Pennsylvania = 245/18440 = 0.0133

p₂ = Proportion of children diagnosed with ASD in Utah = 45/2123 = 0.0212

n₁ = Number of children in Pennsylvania = 18440

n₂ = Number of children in Utah = 2123

x₁ = Number of Children diagnosed with ASD in Pennsylvania = 245

x₂ = Number of Children diagnosed with ASD in Utah = 45

Therefore,

Inference:

Since the calculated Z-test staitistic value of -2.9259 is lesser than the Z-critical value of 2.33, we will not reject the null hypothesis.

Conclusion:

There is no enough evidence to show that the proportion of children diagnosed with ASD in Pennsylvania is more than the proportion in Utah.