Question

2. 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.

(i) Let p₁= proportion of children diagnosed with ASD in Pennsylvania and p₂ = proportion of children diagnosed with ASD in Utah. Which of the following statements correctly defines the null hypothesis H_O?

A.  p₁ + p₂ > 0

B.  p₁ + p₂ = 0

C.  ?₁ = ?₂

D.  p₁ = p₂

Enter letter corresponding to correct answer

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(ii) Let p₁= proportion of children diagnosed with ASD in Pennsylvania and p₂ = proportion of children diagnosed with ASD in Utah. Which of the following statements correctly defines the alternate hypothesis H_A?

A.  p₁ – p₂ < 0

B.  p₁ – p₂ = 0

C.  ?₁ = ?₂

D.  p₁ – p₂> 0

Enter letter corresponding to correct answer

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(iii) Enter the level of significance ? used for this test:

Enter in decimal form. Examples of correctly entered answers:  0.01    0.02    0.05    0.10

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  (iv) Determine pˆPennsylvania and pˆUtah

Enter both in decimal form to nearest ten-thousandth, separated by comma (no spaces) Examples of correctly entered answers:

0.0001,0.0341   

0.0020,0.0500   

0.3000,0.5115

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(v) Find pooled sample proportion p Let pˆPennsylvania = pˆ1 and pˆUtah = pˆ2

Enter in decimal form to nearest ten-thousandth. Examples of correctly entered answers:

0.0001    0.0020     0.0500      0.3000   0.5115

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(vi) Calculate and enter test statistic

Enter value in decimal form rounded to nearest thousandth, with appropriate sign (no spaces). Examples of correctly entered answers:

–2.014      –0.370        +1.600        +11.009

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(vii) Using tables, calculator, or spreadsheet: Determine and enter p-value corresponding to test statistic.

Enter value in decimal form rounded to nearest thousandth. Examples of correctly entered answers:

0.000     0.001     0.030     0.600      0.814 1.000

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(viii) Comparing p-value and ? value, which is the correct decision to make for this hypothesis test?

A. Reject H_o

B. Fail to reject H_o

C. Accept H_o

D. Accept H_A

Enter letter corresponding to correct answer.

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(ix) Select the statement that most correctly interprets the result of this test:

A. The result is not statistically significant at .01 level of significance. Evidence supports the claim that the proportion of children diagnosed with ASD in Pennsylvania is more than the proportion of children diagnosed with ASD in Utah.

B. The result is statistically significant at .01 level of significance. There is not enough evidence to show that the proportion of children diagnosed with ASD in Pennsylvania is more than the proportion of children diagnosed with ASD in Utah.

C. The result is not statistically significant at .01 level of significance. There is not enough evidence to show that the proportion of children diagnosed with ASD in Pennsylvania is more than the proportion of children diagnosed with ASD in Utah.

D. The result is statistically significant at .01 level of significance. Evidence supports the claim that the proportion of children diagnosed with ASD in Pennsylvania is more than the proportion of children diagnosed with ASD in Utah.

Enter letter corresponding to most correct answer

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Answer 1

n1=18440

x1=245

p1= x1/n1= 245/18440= 0.0130

n2= 2123

x2= 45

p2= x2/n2= 45/2123= 0.021196

Two proportion Z test:

a) $H_0:p_1=p_2$

b) $H_a:p_1>p_2$ or $H_a:p_1-p_2>0$

c) Significant level= 0.01

d) P(pennsylvania)= 245/18440= 0.013286

P(Utah)= 45/2123= 0.021196

e) Sample pooled proportion:

$\hat{p}=\frac{X_1+x_2}{n_1+n_2}= \frac{245+45}{18440+2123}=0.014103$

f) The test statistic:

$t= \frac{p_1-p_2}{\sqrt{\hat{p}(1-\hat{p})(1/n_1+1/n_2)}}=\frac{0.013286-0.021196}{\sqrt{0.014103*(1-0.014103)(1/18440+1/2123)}}$

g) P-value: 0.998

h)  Fail to reject Ho

i)The result is not statistically significant at .01 level of significance. There is not enough evidence to show that the proportion of children diagnosed with ASD in Pennsylvania is more than the proportion of children diagnosed with ASD in Utah