Question

Suppose that you are testing the hypotheses Upper H 0​: pequals0.25 vs. Upper H Subscript Upper A​: pnot equals0.25. A sample of size 350 results in a sample proportion of 0.31. ​

a) Construct a 95​% confidence interval for p. ​(Round to three decimal places as​ needed.)

b) Based on the confidence​ interval, can you reject Upper H 0 at alphaequals0.05​? Explain.

​c) What is the difference between the standard error and standard deviation of the sample​ proportion? ​

d) Which is used in computing the confidence​ interval?

Answer 1

a)

p̂ = X / n = 108.5/350 = 0.31
p̂ ± Z(α/2) √( (p * q) / n)
0.31 ± Z(0.05/2) √( (0.31 * 0.69) / 350)
Z(α/2) = Z(0.05/2) = 1.96
Lower Limit = 0.31 - Z(0.05) √( (0.31 * 0.69) / 350) = 0.262
upper Limit = 0.31 + Z(0.05) √( (0.31 * 0.69) / 350) = 0.358
95% Confidence interval is ( 0.262 , 0.358 )

b)

Since 0.25 not contained in confidence interval, reject H0

c)

Standard deviation calculated from population proportion where as standard error is

calculated from sample standard deviation.

d)

Confidence interval calculated using standard error (using sample proportion 0.31)