To test at the 5 level of significance whether the true prop

To test at the 5% level of significance, whether the true proportion of diabetics in the state of Florida between the ages of 18 and 44 years is 25%. It is shown that out of 1,260 diabetics in the state of Florida in 2007, 146 were in the outlined age group. Test the claim that true proportion of diabetics between the ages of 18 and 44 years is actually less than 25%. (round p to four decimal places).

a) Details given in the problem:

b) The number of observations in the sample that are in the outlined age group and not in the outlined age group need to be at least in order to proceed using the normal approximation to binomial.

c) State the Null Hypothesis, Alternative Hypothesis and What tail test is this.

(do not add extra spaces to answers)

Null Hypothesis: vs. Alternative Hypothesis:

What tail test is this? (write only left, right, or both)

d) Find the Critical Statistic (it is recommended you draw a standard normal curve and illustrate this value, give answer to two or three decimal place)

e) Compute the test statistic: (round to two decimal places)

f) Obtain the p-value: (give answer to four decimal places)

g) Conclusion: (only type reject or fail to reject)

h) Using (p, a, z, c) to represent (p-value, alpha, test statistic, critical value) respectively write the statements that support your conclusion for each method: (answers should be inequalities) in other words what are the decision rules that support your conclusion.

P-value method:

Critical value method:

( Please clarify the answer for each one so it\'s easy for me to find it )

Solution

a)

n = 1260
x = 146
alpha = 0.05
p^ = x/n = 146/1260 = 0.115873016 [ANSWER]

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b)

They have to be at least 10.

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c)

Ho: p >= 0.25
Ha: p < 0.25

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d)

As this is a 0.05 level, left tailed z test, then

zcrit = -1.645 [ANSWER]

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e)

Formulating the null and alternatuve hypotheses,          
          
Ho:   p   >=   0.25
Ha:   p   <   0.25

As we see, the hypothesized po =   0.25      

Getting the point estimate of p, p^,          
          
p^ = x / n =    0.115873016      
          
Getting the standard error of p^, sp,          
          
sp = sqrt[po (1 - po)/n] =    0.012198751      
          
Getting the z statistic,          
          
z = (p^ - po)/sp =    -10.99514082   [ANSWER]

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F)
  
          
As this is a    1   tailed test, then, getting the p value,  
          
p =    2.01642*10^-28   [ANSWER]  

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g)

As P < 0.05, we REJECT THE NULL HYPOTHESIS.

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h)

As P = 2.01642*10^-28 < 0.05, and z = 10.995 < -1.645, we REJECT HO.

Thus, there is significant evidence at 0.05 level that that true proportion of diabetics between the ages of 18 and 44 years is actually less than 25%. [CONCLUSION]