Question

The diameters of apples from a certain farm follow the normal distribution with mean 4 inches and standard deviation 0.4 inch. Apples can be size-sorted by being made to roll over mesh screens. First the apples are rolled over a screen with mesh size 3.5 inches. This separates out all the apples with diameters less than 3.5 inches. Second, the remaining apples are rolled over a screen with mesh size 4.3 inches. Find the proportion of apples with diameters less than 3.5 inches.

• Round your answer to four decimal places.

Answer 1

Solution:

Given: The diameters of apples from a certain farm follow the normal distribution with mean 4 inches and standard deviation 0.4 inch.

Thus X ~ Normal

(u = 4,0 = 0.4)

We have to find: the proportion of apples with diameters less than 3.5 inches.

P( X < 3.5 )= ...............?

Find z score for x = 3.5

1 -

3.5-4 -0.5 0.4 -= -1.25

Thus we get:

P( X < 3.5 )= P( Z < -1.25 )

Look in z table for z = -1.2 and 0.05 and find corresponding area.

.07 .08 .09 | 1: -3.4 -3.3 -3.2 -3.1 -3.0 -2.9 -2.8 -2.7 -2.5 -2.5 -2.4 -2.3 -2. 2 -2.1 -2.0 -19 -1.8 -1.7 -1.6 -1.5 -1.4 -13 -1.2 -1.1 00 .01 .02 ,03 ,04 ,05 106 .0003 ,0003 0003 .000 .0003 .0 .0003 ,0005 .0005 .0005 ,0004 .000 .09 .0004 .0007 ,0007 0006 .0006 .0006 . 0 6 .0006 0010 .0009 ,0009 ,0009 ,0008 ,000 ,0008 .0013 .0013 .0013 0012 .0012 0411 .0011 .0019 ,0018 0018 .0017 0016 .0016 .0015 .0026 .0025 .0024 .0023 .0023 1.0021 .0035 .0034 0033 0032 .01 .00 0 0029 0047 .0045 .004 .043 10041 .0 0 .0039 ,0062 0060 .0059 .007 1.0055 54 20052 .0062 ,0080 .0078 .075 .00730 F1 .0069 .0107 ,010 .0102 ,0099 .0096 094 0091 0139 .0136 20132 .0129 .0125 .00 22 20119 .0179 ,0174 20170 .0166 .0162 058 20154 .0228 .0222 0217 .0212 .0207 .002 2017 0287 .0281 0274 .0268 .0262 .056 10250 40359 ,0351 .034 .0336 .0329 .0314 0446 .0436 ,0427 .0418 .0409 .4$1 ,0392 .0546 ,0537 1.0526 .0516 .0505 15 .0485 0668 .0655 0643 .0630 .0618 ,0006 0594 0606 ,0793 .0778 .0764 .09 . 05.0721 .0968 .0951 0934 .0918 0901 5 10868 11111111112143- .1038 .1357 .1335 1314 192 .1271 .1251 .1230 .0003 .0003 ,0000 ,0004 .0005 .0005 000B 10007 .0011 .0010 0015 .0014 .0021 .0020 0028 007 .00238 .0037 20051 009 20068 .006 20089 .007 01160113 20150 .016 .0192 .01 .024 .029 .03070301 0384 0375 .0475 .0465 0582 .0571 .0708 .0694 .0853 ,0838 .1020 .1003 .1210 .1190 .0002 | ,0003 .0005 | 0007 .0010 | 0014 20019 20026 .0036 .0048 .0064 0084, .0110 | .0143 .01831 .0233 | .0294 .0367 .0455 | .0559 | .0681 | ,0823 .0985 | .1170 |

P( Z < -1.25 )= 0.1056

thus

P( X < 3.5 )= P( Z < -1.25 )

P( X < 3.5 )= 0.1056

Thus the proportion of apples with diameters less than 3.5 inches is 0.1056