Suppose that you are testing the hypotheses Ho p 040 vs HA

Suppose that you are testing the hypotheses Ho: p = 0.40 vs HA: p> 0.40. A sample size of 200 results in a sample proportion of 0.55. a) Construct a 90% confidence interval for p. b) Based on the confidence interval, at p-level =.05 can you reject Ho? Explain. c) What is the difference between that standard error and standard deviation of the sample proportion? d) Which is used in computing the confidence interval?

Solution

a)

Note that              
              
p^ = point estimate of the population proportion = x / n =    0.55          
              
Also, we get the standard error of p, sp:              
              
sp = sqrt[p^ (1 - p^) / n] =    0.035178118          
              
Now, for the critical z,              
alpha/2 =   0.05          
Thus, z(alpha/2) =    1.644853627          
Thus,              
Margin of error = z(alpha/2)*sp =    0.057862855          
lower bound = p^ - z(alpha/2) * sp =   0.492137145          
upper bound = p^ + z(alpha/2) * sp =    0.607862855          
              
Thus, the confidence interval is              
              
(   0.492137145   ,   0.607862855   ) [ANSWER]

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b)

YES, because the left tail of this confidence interval is greater than 0.40 (it is 0.4921). [ANSWER, YES]

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c)

The standard deviation is independent on sample size, the standard error is dependent on sample size.

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d)

It is the standard error that is used.