Question

Consider the following hypotheses. Ho p 0.42 H1 : p #0.42 Given that p 0.36, n200, and a 0.05, answer the following questions a. What conclusion should be drawn? b. Determine the p-value for this test. a. Determine the critical value(s) of the test statistic. (Use a comma to separate answers as needed. Round to two decimal places as needed.) Calculate the test statistic. Zo = (Round to two decimal places as needed.) Choose the correct answer below A. Reject Ho-There is sufficient evidence that p # 0.42. B. Do not reject Ho . There is insufficient evidence that p # 042. ° C. Do not reject Ho . There is sufficient evidence that p # 0.42. ( D. Reject Ho. There is insufficient evidence that p # 0.42. b. p-value(Round to three decimal places as needed.)

Answer 1

Solution:

Given: Hypothesis:

$H_{0}: p = 0.42$

$H_{1}: p \neq 0.42$

Sample proportion = $\bar{p}=0.36$

Sample size = n = 200

Level of significance = $\alpha = 0.05$

Part a)

Find z critical value(s)

Since $H_{1}: p \neq 0.42$ is not equal to type, that is non-directional , thus this is two tailed test.

Thus we find two z critical values.

Find an Area = $\alpha / 2 = 0.05 /2= 0.025$

Look in z table for area = 0.0250 or its closest area and find z value

.02 07 08 .00 .01 .03 04 .05 .06 .09 3.4 .0003 .0003 .0003 .0003 .0003 .0003 .00р3 .0003 .0003 .0002 3.3 .0005 0005 .0005 .0004 .0004 0004 .0004 .0004 0004 0003 3.2 .0007 0007 .0006 .0006 .0006 .0006 .0006 0005 .0005 .0005 -3.1 .0010 .0009 .0009 .0009 .0008 .0008 .0ф8 .0008 .0007 .0007 3.0 .0013 .0013 .0013 .0012 .0012 .0011 .0011 .0011 .0010 .0010 2.9 .0019 .0018 .0018 .0017 .0016 .0016 .00 .0015 0014.0014 2.8 .0026 .0025 .0024 0023 .0023 .0022 .0021 .0021 0020 .0019 -2.7 .0035 .0034 .0033 0032 0031 0030 0029 .0028 .0027 0026 2.6 .0047 0045 .0044 .0043 .0041 .0040 .0019.0038.0037.0036 2.5 .0062 .0060 .0059 0057 0055 0054 .002 .0051 .0049 .0048 2.4 .0082 .0080 .0078 .0075 .0073 .0071 .00#9 .0068 .0066 .0064 0099 0096 0094 .0091 .0089 0087 0084 -2.3 .0107 .0104 .0102 .0099 .0096 .0094 .00р1 .0089 .0087 .0084 2.2 .0139 .0136 .0132 .0129 .0125 .0122 .0119 .0116 .0113 .0110 0166 .0162 .0158 .0164 .0150 0146 .0143 0217 .0212 .0207 .0202 017 .0192 0188 .0183 1.9 0207281 U274026002620256.0250 0244 0239 .0233 1.8 .0359 .0351 .0344 .0336 .0329 0322 .0314 .0307 .0301 .0294 -2.1 .0179 .0174 .0170 . 2.0 .0228 .0222 数SI 0212

Area 0.0250 corresponds to -1.9 and 0.06

thus z = -1.96

this is z critical value for left tailed 0.0250 area

similarly z critical value for right tailed 0.0250 area would be same but positive, that is: z = 1.96

thus z critical values are: ( -1.96 , 1.96 )

Calculate test statistic:

$z= \frac{\bar{p}-p}{\sqrt{\frac{p\times (1-p)}{n}}}$

$z= \frac{0.36-0.42}{\sqrt{\frac{0.42\times (1-0.42)}{200}}}$

$z= \frac{-0.06 }{\sqrt{\frac{0.42\times 0.58}{200}}}$

$z= \frac{-0.06 }{\sqrt{0.001218 }}$

$z= \frac{-0.06 }{0.0349 }$

$z=-1.72$

Choose the correct answer:

Since z test statistic = -1.72 is neither < -1.96 , nor greater than 1.96, we do not reject H0. That is p = 0.42 is true.

thus correct answer is:

B. Do not reject H0. There is insufficient evidence that $p \neq 0.42$

Part b) Find p-value:

p-value = 2 X P( Z < z test statistic value)

p-value= 2 X P( Z < -1.72)

Thus look in z table for z = -1.7 and 0.02 and find area.

Thus from z table we get:

P( Z < -1.72)= 0.0427

Thus

p-value= 2 X P( Z < -1.72)

p-value= 2 X 0.0427

p-value = 0.0854

p-value= 0.085