Question

1. Given:Ho:p≥0.4,Ha:p<0.4,n=25,RejectHo ifX≤7
   (a) Find the level of significance .
   (b) If in fact p = 0.3, what would be the probability of making a type II error?
   (c) If the true value of p were 0.3, find the probability the test would detect that this is the case.

   What is this probability called?
   (d) Suppose that X is observed to be xo = 5.

   (i) What is your decision? (ii) What type of error are you subject to? (iii)What is the P-value?

   (e) By hand draw the power curve for this hypothesis test.
   Include the table giving the values of K(p) for p = 0.05, 0.10, 0.20, ..., 0.90, 0.95 and indicate where  is on your curve.

   (f) For the hypotheses, Ho: p ≥ 0.4, Ha: p < 0.4, n = 25, set up a rejection region so that  is as close as possible to, but does not exceed 0.10. State both the 'nominal'  and the 'exact' .

Answer 1

(a)

Standard error of proportion, SE = = = 0.09798

Reject H0 if X 7 or p 7/25 = 0.28

= P(Reject H0 | H0 is True) = P(p 0.28 | p = 0.4)

= P[Z (0.28 - 0.4)/0.09798]

= P[Z -1.22]

= 0.1112

(b)

Standard error of proportion, SE = = = 0.09165151

Probability of making a type II error = P(Fail to reject H0 | H0 is False) = P(p > 0.28 | p = 0.3)

= P[Z > (0.28 - 0.3)/0.09165151]

= P[Z > -0.22]

= 0.5871

(c)

Probability the test would detect that this is the case = 1 - P(Reject H0 | H0 is False) = 1 - 0.5871 = 0.4129

This probability called is power of the test.

(d)

(i)

Since x0 = 5 is less than the critical value of 7, we reject H0.

(ii)

Since, we reject H0, we may commit Type I error in case null hypothesis H0 is true.

(iii)

p = x0 / n = 5/25 = 0.2

Test statistic, z = (0.2 - 0.4) / 0.09798 = -2.04

P-value = P(z < -2.04) = 0.0207

(e)

For, p = 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95

We will calculate the

Standard error of proportion, SE =    and

Power = P(Reject H0 | H0 is False) = P(p < 0.28 | p = given value)

The Power curve is,

(f)

For = 0.10, the critical value of z is -1.28

Critical value of p = 0.4 - 1.28 * 0.09798 = 0.2746

Critical value of X = 25 * 0.2746 = 6.865 6 (Round to previous integer. Note if we will use X = 7, then will exceed 0.10)

Reject H0 if X 6

p = 6/25 = 0.24

= P(Reject H0 | H0 is True) = P(p 0.24 | p = 0.4)

= P[Z (0.24 - 0.4)/0.09798]

= P[Z -1.63]

= 0.0516

For this rejection region, nominal = 0.05 and exact = 0.0516