Question

Suppose that the distance of fly balls hit to the outfield (in baseball) is normally distributed with a mean of 229 feet and a standard deviation of 44 feet. We randomly sample 42 fly balls.

Let   X¯=   average distance in feet for 42 fly balls.

Enter numbers as integers or fractions in "p/q" form, or as decimals accurate to nearest   0.01 .

1. (.20)  X¯∼    (pick one) PBNE    (  , ) .
2. (.20)   Use the mean and standard deviation of   X¯   to determine the   z   value for   X¯=240 .

3. (.20)   Illustrate   P(X¯>240)   as an area by adjusting the slider along the horizontal axis to control the   z   value.

4. (.20)   Illustrate   P(X¯>240)   as an area by adjusting the slider along the horizontal axis to control the   z   value.

   Fill left    Fill right ??

5. (.20)   What is the probability that the 42 balls traveled an average of greater than   240   feet?

   P(X¯>240)=  

6. (.20)   Find the 80th percentile of the distribution of the average of   42   fly balls. That is, find   x   so that   P(X¯

   x=   

Answer 1

Given that Population mean =229 ft and standard deviation =44 and no of samples selected are 42, n=42 also the distribution is normal

now,

a) $\bar{X}\sim 229$ Since it is normally distributed hence by using central limit theorem.

b) Now at X(bar), Sample mean =240 using Z formula

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

Z =1.62

c) Using Z table to find Probability P(X¯>240)

=0.0526

d) For area P(X¯>240) the curve shows below

e) Now at X(bar), Sample mean =240 using Z formula

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

Z =1.62

Using Z table to find Probability P(X¯>240)

=0.0526

f) At 80th percentile the Z value taken from Z table as 0.845

hence using the Z formula again

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

X= 234.737

The Z table used is shown below

Table of Standard Normal Probabilities for Negative Z-scores Table of Standard Normal Probabilities for Positive 7-scores 3.4 0.0003 0,0003 0.0003 0,0003 ,0003 00003 00003 0.0003 0.0003 0.0002 ฐี.OİD.CO I 3 0.001 3 ú.DCDI 3 U.0012 0.0012 ข.001 1 0.0011 0.0011 0.0010 D.C. -2.9 0.0019 0.0018 0.0018 00017 0,0016 0.0016 00015 0.0015 0.001 0.0014 0.0026 0.D05 .4 0.023 O23 0.0022 0002 002 0.0020 D.019 -2.7 0.0035 0.0034 0.0033 0.0032 0.003 0.030 00029 00028 0.0027 0.0026 26 0.07 0.0015 0.0011 00013 0,00 0.0010 00039 0.0038 0.0037 0.0036 0.6554 0.659 0.662 0.6664 06700 0.ST36 D672 D.6308 0.6844 0.6379 0.6915 0.6950 0,6985 0.7019 0.705 0.7088 07123 0.7157 0,7190 0.7224 0.7257 0729 0.7324 0.7357 07389 07422 7454 D.7486 0.7517 0.7549 0.7580 0.71 0.7642 0.673 0.770 07734 07764 0.79 0.7823 0.7852 0.7881 0.7910 0.7939 0.97 0.7995 0.8023 08051 08078 0.8106 0.8133 O.8 I 59 0.S 186 0.8212 0.823民 0.8264 0.3289 08315 D.S340 D.8365 0.8389 0.8413 0.8438 0816 08485 .8508 0853 8554 D577 0.8599 0.8621 0.5 2.4 0.0082 0.0080 0.0078 0.0075 0003 0.00 00059 0.0068 0.0066 0.0064 23 0.07 .0104 .0102 og .0D6,4 000 00 0.008 D.0084 -2.2 0.0139 0.0136 0.0132 .0129 0.0125 00122 00119 .0116 0.013 0.0110 -2.1 0.0179 0.0174 0.010 0.0166 0.0162 0.0158 00154 00150 0.0146 0.0143 -2.0 | 0.0228 0.0222 ถ.ถ217 0.02 12 ด.0207 ด.0202 0.0197 0.DI 92 0.Ol88 D.D183 1.3 0.9032 0.9049 0.9066 0.9082 0.9099 09115 0913 09147 0.9162 0.9177 0.9102 ด.9207 0.9722 0.9236 0.9251 09265 09279 D9292 0.9306 0.93เจ 1.5 0.9332 0.9345 0.935 .930 0.9382 0939 09406 09418 0.9429 0.9441 n9452 0.9463 0.0474 ด.9484 09495 09505 09515 D9525 ถ9535 ถ.9545 0,0359 0.D351 ถ.กา44 0.0336 0.0379 0.0322 003 1 4 0.0307 0.0301 0.0204 -1.7 | 0.0446 0.0476 0.0427 0.0418 0.0409 0.040 l ao392 0.0384 0.0375 D.D367 160.0548 0.053 0.026 0.0516 00S0 00493 00485 005 0.065 00455 -1,5 | 0,0668 0.0655 0.0543 0.0630 0.0618 ด.0606 00594 00582 00571 D0559 -1.4 0.0808 0.0793 U.U778 U.0?64 U.U 749 0.0733 00721 0008 00694 D.0681 -1.3 | 0.0968 0.0951 U.0934 ข.0918 U.0901 U.0885 00869 00853 0.0838 D.U823 -1.2 | 0.1151 0.1131 о,1112 0.1093 0.1075 0.1056 0.1038 0.1020 0.1003 00985 -1.1 0.1357 D. 1335 U. 1314 U.1292 U.1271 ข. 1 251 0.1230 0.1210 0.1190 D.1170 10 0.1587 52 0.1539 1515 0.1492 0.19 0.1446 01423 0.140 0.1379 09 0.1841 0.1814 0.1788 .1762 0.1736 7 01685 01660 D.1635 D.1611 0.9641 0.949 0.956 .9664 096 9678 9686 D.963 09699 0906 1.9 09713 0.97 19 0.9726 ด.9732 00738 0.9744 09750 D 9756 09761 00767 U.9T2 0977 0.9783 0.988 09793 9T9 0903 9812 017 2.1 0.982 0.26 0830 0.9834 09838 09842 09846 .9850 09854 857 ถ 9861 ถ.9864 0.9868 ด.987' 09875 09878 09881 09884 09887 ถ.9890 2.3 U.9893 0.9896 0.9898 0.990 1 0.9904 0g906 0gX9 D.9911 U.ggl3 0.gg 16 0.9918 09920 0,9922 09925 09927 09929 09931 09932 0.993 0,9936 07 0.2420 0.2389 0.2358 0.2327 0,2296 0.2266 02236 0.2206 0.2177 0.2148 0.2743 0.2709 0.2676 d.2643 0.2611 ข.2578 02546 0.514 D.2483 D.245 1 0.9074 0.TS 0.9976 . 0977 09978 D9979 9979 0.9980 0,998 2.9 0.998 0.9982 0992 0,9983 0.9981 0998 09985 0.9985 0.9986 0,9986 -0.4 | 0.3446 0.3409 0.3372 0.3336 0.3300 0.3264 O.3228 0.3192 0.3156 0.3121 -Q3 |0.3821 0.3783 0.3745 d.3707 C.3669 0.3632 0.3594 03ฐ57 0.3520 D.3483 0.2 04207 0.4168 0.4129 0.4090 0.4052 0.4013 0397 03936 0.3897 0.3859 al 0.4602 0.4562 0.4522 ต.4483 0.4443 0.4404 0.4364 04325 0.4286 D.4247 -0.0 | 0.5000 D.4960 0.4920 d.4gg0 0.4840 0.4501 O.4761 0,4721 D.466 1 D.4641 Note that the probabilities given in this table represent the area to the LEFT of the z-score The area to the RIGHT of a z-score 1-the area to the LEFI, of the z-score