Question

You wish to test the following claim ( H a ) at a significance level of α = 0.002 . H o : p = 0.46 H a : p < 0.46 You obtain a sample of size n = 295 in which there are 113 successful observations. What is the test statistic for this sample? (Report answer accurate to two decimal places.) test statistic = What is the p-value for this sample? (Report answer accurate to four decimal places.) p-value = The p-value is... less than (or equal to) α greater than α This test statistic leads to a decision to... reject the null accept the null fail to reject the null As such, the final conclusion is that... There is sufficient evidence to warrant rejection of the claim that the population proportion is less than 0.46. There is not sufficient evidence to warrant rejection of the claim that the population proportion is less than 0.46. The sample data support the claim that the population proportion is less than 0.46. There is not sufficient sample evidence to support the claim that the population proportion is less than 0.46

Answer 1

Solution:

Given:

Level of significance =  α = 0.002

Hypothesis of the study are:

H_o : p = 0.46 Vs H_a : p < 0.46

Alternative hypothesis H_a is the claim.

Sample size = n = 295

x = 113

Part a) . What is the test statistic for this sample?

z test stat for proportion

-P px(1-P)

where

11.3 095 = 0.3831

thus

-P px(1-P)

0.3831 – 0.46 0.46x(1-0.46) 295

-0.0769 0.46x0.54 295

-0.0769 0.00084203

-0.0769 10.0290178

2 = -2.65

Part b) What is the p-value for this sample?

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

p-value = P( Z< -2.65)

Look in z table for z = -2.6 and 0.05 and find corresponding area.

.08 0003 7 1-3.4 -3.3 1-3.2 1-3.1 -3.0 12.9 -2.8 -2.7 -2.6 ,00 ,0003 ,0005 .0007 .00 10 .0013 .0019 .0026 ,0035 A047 .01 0003 .0005 .0007 ,0009 .0013 .0018 .0025 .0034 -40 .02 .03 .04 .05 .06 .07 ,0003 ,0003 .0003,0003,0003,0003 .0005 ,0004 .0004 ,0004 .0004 .0006 | 10006 .0006 .0006 .0006 ,0005 ,0000 ,0009 ,0008 .0018 ,0008 ,0008 .0013 .00 1 .0011 .0011 .0018 .0017 .0016 .0016 .0015 .0015 .0024 .0023 .0023 .00*2 .0021 .0021 .0033 .0032 .0031 .0050 .0029 .0028 .00 43 .UUM .0040 .0039 ,0038 .0005 ,0007 .0010 .0014 .0020 .0027 .0037 .09 0002 10003 .0005 ,0007 ,0010 .0014 .0019 .0026 ,0036

P( Z< -2.65) = 0.0040

Thus

p-value = P( Z< -2.65)

p-value = 0.0040

The p-value is. 0.0040 greater than α = 0.002

This test statistic leads to a decision to fail to reject the null.

As such, the final conclusion is that : There is not sufficient sample evidence to support the claim that the population proportion is less than 0.46