Question

You wish to test the following claim (H) at a significance level of a = 0.002. H: = 76.8 Hau < 76.8 You believe the population is normally distributed and you know the population standard deviation is o = 12.5. You obtain a sample mean of ä = 71.6 for a sample of size n = 21. a. What is the test statistic for this sample? test statistic - Round to 3 decimal places b. What is the p-value for this sample? P-value = Round to 4 decimal places.

Answer 1

We have hypothesis as

$H_0:\mu = 76.8$

Ha: Η 76.8

Given information

$\bar x = 71.6$

o = 12.5

n = 21

The formula of test statistics

$Z = \frac{\bar x - \mu}{\frac{\sigma}{\sqrt{n}}}$

$Z = \frac{71.6 - 76.8}{\frac{12.5}{\sqrt{21}}}$

$Z = \frac{-5.2}{\frac{12.5}{4.582575695}}$

$Z = \frac{-5.2}{2.727723628}$

We round the value of test statistics up to 3 decimal place

Test statistics = Z = -1.906

Our test is left tailed because H_a hypothesis has < sign

To find the p-value we use Z table

Because the Z table always gives the left side probability.

Test statistics = Z = -1.906

Round the value of test statistics up to 2 decimal place

Test statistics = Z = -1.91

We look for row headed -1.9 and column headed 0.01

Because -1.91 = -1.9 + ( -0.01)

TABLE 5 Areas of a Standard Normal Distribution (a) Table of Areas to the Left of z z .00 .01 .02 .03 .04 .05 .06 .07 .08 .09 .0003 .0004 .0005 .0008 .0011 .0015 .0021 .0028 .0038 .0051 .0068 -3.4 -3.3 -3.2 -3.1 -3.0 -2.9 -2.8 -2.7 -2.6 -2.5 -2.4 -2.3 -2.2 -2.1 -2.0 - 1.9 -1.8 -1.7 -1.6 -1.5 -1.4 -1.3 – 1.2 .0089 .0116 .0150 .0192 .0003 .0003 .0003 .0003 .0003 .0003 .0003 .0005 .0005 .0005 .0004 .0004 .0004 .0004 .0007 .0007 .0006 .0006 .0006 .0006 .0006 .0010 .0009 .0009 .0009.0008 .0008 .0008 .0013 .0013 .0013 .0012.0012 0011 .0011 .0019 .0018 .0018 .0017 .0016 .0016 .0015 .0026 .0025 .0024 .0023 .0023 .0022 .0021 .0035 .0034 .0033 .0032.0031 .0030 .0029 .0047 .0045 .0044 .0043 .0041 .0040 .0039 .0062 .0060 .0059 .0057 .0055 .0054 .0052 .0082 .0080 .0078 .0075 .0073 .0071 .0069 .0107 .0104 .0102 .0099 .0096 .0094 .0091 .0139 .0136 .0132 .0129 .0125 .0122 .0119 .0179 .0174 .0170 .0166 .0162 0158 .0154 .0228 .0222 .0217 .0212 .0207 0202 .0197 .0287 .0281|1.0274 .0268 .0262 0256 .0250 .0359 .0351 .0344 .0336 .0329 .0322 .0314 .0446 .0436 .0427 .0418 .0409 .0401 .0392 .0548 .0537 .0526 .0516 .0505 .0495 .0485 .0668 .0655 .0643 .0630 .0618 0606 .0594 .0808 .0793 .0778 .0764 0749.0735 .0721 .0968 .0951 .0934 .0918 .0901 .0885 .0869 .1151 .1131 .1112 .1093 .1075 1056 .1038 .1357 .1335 .1314 1292 1271 .1251 . 1230 .1587 .1562 .15391515 .1492 .1469 .1446 .1841 .1814 .1788 1762 1736 1711 1685 .2119 .2090 .2061 .2033 .2005 .1977 .1949 .2420 .2389 .2358 .2327 .2296 .2266 .2236 .2743 .2709 .2676 .2643 .2611 .2578 .2546 .3085 .3050 .3015 .2981 .2946 .2912 .2877 .3446 .3409 .3372 .3336 .3300 .3264 3228 .3821 .3783 .3745 .3707 3669 .3632 3594 4207 4168 4129 4090 4052 4013 .3974 4602 4562 4522 4483 .4443 .4404 4364 .5000 4960 4920 4880 4840 .4801 4761 .0003 .0002 .0004 .0003 .0005 .0005 .0007 .0007 .0010 .0010 .0014 .0014 .0020 .0019 .0027 .0026 .0037 .0036 .0049 .0048 .0066 .0064 .0087 .0084 .0113 .0110 .0146 .0143 .0188 .0183 .0239 .0233 .0301 .0294 .0375 .0367 .0465 .0455 .0571 .0559 .0694 .0681 .0838 .0823 .1003 .0985 .0244 .0307 .0384 .0475 .0582 .0708 .0853 .1020 .1210 - 1.1 1190 1170 -1.0 .1423 .1660 .1922 .2206 .2514 .2843 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 -0.0 .1401 1379 .1635 .1611 .1894 .1867 .2177 .2148 .2483 .2451 .2810 .2776 .3156 .3121 .3520 .3483 .3897 .3859 .4286 4247 4681 4641 .3192 .3557 .3936 4325 4721 For values of z less than - 3.49, use 0.000 to approximate the area.

From table

p-value = 0.0281

Final answer:-

a) Test statistics = -1.906

b) p-value = 0.0281