Question

Find the p -value for the hypothesis test. A random sample of size 52 is taken. The sample has a mean of 388 and a standard deviation of 85. H0: µ = 400 Ha: µ< 400 The p -value for the hypothesis test is?

Answer 1

Solution:

Given:

Sample size = n= 52

Sample mean = $\bar{x} = 388$

Sample standard deviation = s = 85

H0: µ = 400 Ha: µ< 400

We have to find the p -value for the hypothesis test.

Since sample size =n =52 is large ,we will use z test statistic and sample standard deviation as an estimate of population standard deviation

thus find z test statistic:
$z=\frac{\bar{x}-\mu }{\sigma /\sqrt{n}}$

$z=\frac{ 388 - 400 }{85 /\sqrt{52}}$

$z=\frac{ -12 }{85 /7.211103 }$

$z=\frac{ -12 }{11.7873792 }$

$z=-1.02$

Thus p-value is:

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

p-value = P(Z < -1.02)

Look in z table for z = -1.0 and 0.02 and find corresponding area.

.00 01 .02 .03 .04 .05 .06 07 .08 .09 z .0003 .0005 .0006 .0009 .0013 .0018 .0024 .0033 .0044 .0059 .0078 .0102 .0 32 0003 0003 .0004 .0003 .0003 .0003 0003 .0003 .0002 -3.4 0003 .0004 .0004 .0003 -3.3 .0005 .0005 .0004 0004 .0004 0007 0010 0005 .0007 0005 -3.2 .0007 0006 .0006 .0006 0006 0005 .0009 .0008 .0008 .0008 .0007 -3.1 .0009 0008 -3.0 0013 .0013 .0012 .0012 .0011 .0016 0011 0011 .0010 .0010 0015 .0018 .0017 .0016 .0015 0021 .0014 .0014 -2.9 0019 .0020 0019 -2.8 0026 0035 .0025 .0023 .0023 0022 .0030 .0021 .0034 -2.7 .0032 .0031 .0029 0028 0038 .0027 .0026 .0041 .0040 0039 .0037 .0036 -2.6 0047 .0045 .0043 -2.5 .0062 .0060 0057 .0055 .0054 .0052 .0051 .0049 .0048 .0080 .0075 .0073 .0071 .0069 .0068 .0089 0066 .0064 -2,4 0082 .0107 .0104 .0094 .0087 .0084 -2.3 .0099 0096 0091 .0119 -2.2 0139 .0136 ,0129 0125 .0122 0116 .0113 0110 ,0166 .0150 .0143 -2.1 -2.0 .0170 .0217 274 0344 0427 0526 0643 0778 0934 1112 1314 0162 .0158 0154 .0146 0179 .0174 0188 .0239 .0222 0207 0262 .0228 .0212 .0202 .0197 0192 .0183 0233 .0287 0281 .0268 .0256 0250 0244 -1.9 .0307 .0301 0294 -1.8 0359 .0351 0336 .0329 0322 0314 .0446 .0436 .0392 -1.7 .0418 .0409 .0401 .0384 .0375 .0367 0516 .0505 .0495 0485 0475 0465 0455 -1.6 0548 .0537 .0582 .0559 -1.5 0668 .0655 .0630 .0618 .0606 .0594 .0571 0793 .0951 0808 0764 0749 .0735 0721 .0708 .0694 0681 -1.4 .0823 0985 .0853 .0838 -1.3 .0968 .0918 .0901 .0885 0869 -1.2 1003 .1151 .1131 .1093 .1075 .1056 1038 1230 1020 1335 1292 1271 1251 1210 .1190 .1170 -1.1 1357 -1.0 1507 1562 1539 1515 1492 1469 1446 .1423 .1401 .1379

Thus

P( Z < -1.02 ) = 0.1539

thus

p-value = P(Z < -1.02)

p-value = 0.1539