Question

To test ?0: p = 0.4; ?1: p ≠ 0.4, a simple random sample of size n = 1000 is obtained from a population such that ? ≤ 0.05?. (a) If x = 420 and n = 1000, compute the test statistic 0 z . (b) Test the hypothesis using (i) the classical approach and (ii) the P-value approach. Assume an ? = 0.01 level of significance. (c) What is the conclusion of the hypothesis test?

Answer 1

Solution:

Given:

?0: p = 0.4; ?1: p ≠ 0.4

n = 1000

x = 420

Part a) Compute the test statistic z

where

thud

Part b) Test the hypothesis using

(i) the classical approach:

level of significance = ? = 0.01

Since this is two tailed test , find

Look in z table for Area = 0.0050 and find corresponding z value.

Area 0.0050 is in between 0.0049 and 0.0051, and both the area are at same distance from 0.005

thus we look for both area and find both z values.

Area 0.0049 corresponds to -2.5 and 0.08 , thus z= -2.58

Area 0.0051 corresponds to -2.5 and 0.07 , thus z= -2.57

Thus average of both z values is = ( -2.57 + -2.58 ) / 2 = -2.575

Thus critical z value is = -2.575

Since this is two tailed test , there are two z critical values = ( -2.575 , 2.575 )

Decision Rule:
Reject null hypothesis ,if z test statistic value < z critical value=-2.575 or z test statistic value > z critical value=2.575 , otherwise we fail to reject H0.

Since z test statistic value = is neither < -2.575 , nor > 2.575, that is it does not fall in rejection region, we fail to reject H0.

ii) the P-value approach.

i) For right tailed test , p-value is:

p-value = P(Z > z test statistic)

ii) For left tailed test , p-value is:

p-value = P(Z < z test statistic)

iii) For two tailed test , p-value is:

p-value = 2* P(Z > z test statistic) if z is positive

p-value = 2* P(Z < z test statistic) if z is negative

Since this is two tailed test and z is positive, we use:

p-value = 2* P(Z > z test statistic)  

p-value = 2* P(Z > 1.29)

p-value = 2* [ 1 - P(Z < 1.29) ]

Look in z table for z = 1.2 and 0.09 and find corresponding area.

P( Z< 1.29) = 0.9015

thus

p-value = 2* [ 1 - P(Z < 1.29) ]

p-value = 2* [ 1 - 0.9015 ]

p-value = 2* 0.0985

p-value = 0.1970

Decision Rule:
Reject null hypothesis H0, if P-value < 0.01 level of significance, otherwise we fail to reject H0

Since p-value = 0.1970 > 0.01 level of significance, we fail to reject H0

Part c) What is the conclusion of the hypothesis test?

At 0.01 level of significance, we do not have sufficient evidence to reject the null hypothesis that p =0.4

that is : population proportion is not different from 0.4