A random sample of women is obtained and each person in the

A random sample of women is obtained, and each person in the sample is asked if she would purchase a new shoe model. To determine if the new shoe model would have sales of at least 25% to meet corporate profit objectives, the following hypothesis test is performed at a level of = 0.03 using p hat as the sample proportion of women who said yes. What value of the sample proportion, , is required to reject the null hypothesis

given the following sample sizes? a) n = 400 b) n = 225 c) n = 625 d) n = 900

Solution

As we can see, right is a right tailed test from the word \"at least\".

Thus, for a right tailed test, the critical z is, at 0.03 level,

zcrit = 1.880793608

Thus, as

zcrit = (pcrit - po) / sp

pcrit = po + zcrit*sp

and

sp = sqrt(po (1-po) / n).

a)

If n = 400,

sp = sqrt(0.25*(1-0.25)/400) = 0.021650635

Then

pcrit = po + zcrit*sp = 0.25 + 1.880793608*0.021650635 = 0.290720376 [ANSWER]

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b)

If n = 225,

sp = sqrt(0.25*(1-0.25)/225) = 0.028867513

Then

pcrit = po + zcrit*sp = 0.25 + 1.880793608*0.028867513 = 0.304293834 [ANSWER]

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c)

If n = 625,

sp = sqrt(0.25*(1-0.25)/625) = 0.017320508

Then

pcrit = po + zcrit*sp = 0.25 + 1.880793608*0.017320508 = 0.282576301 [ANSWER]

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d)

If n = 900,

sp = sqrt(0.25*(1-0.25)/900) = 0.014433757

Then

pcrit = po + zcrit*sp = 0.25 + 1.880793608*0.014433757 = 0.277146918 [ANSWER]