Question

At a tennis tournament a statistician kept track of every serve. The statistician reported that the mean serve speed of a particular player was 97 miles per hour (mph) and the standard deviation of the serve speeds was 11 mph. Assume that the statistician also gave us the information that the distribution of the serve speeds was bell shaped. What proportion of the playerʹs serves are expected to be between 119 mph and 130 mph?

The proportion is ____ (round to 4 decimal places as needed)

Answer 1

ANSWER:

Given that,

At a tennis tournament a statistician kept track of every serve. The statistician reported that the mean serve speed of a particular player was 97 miles per hour (mph) and the standard deviation of the serve speeds was 11 mph. Assume that the statistician also gave us the information that the distribution of the serve speeds was bell shaped. What proportion of the playerʹs serves are expected to be between 119 mph and 130 mph?

Mean = $\mu$ = 97 mph

Standard deviation = $\sigma$ = 11 mph

z = (X-$\mu$)/$\sigma$ = (X-97)/11

P(119 < X < 130) = P((119-97)/11 < (X-$\mu$)/$\sigma$ < (130-97)/11)

P(119 < X < 130) = P(22/11 < Z < 33/11)

P(119 < X < 130) = P(2 < Z < 3)

P(119 < X < 130) = P(Z < 3)-P(Z < 2)

P(119 < X < 130) = 0.99865-0.97725 (From z score table)

P(119 < X < 130) = 0.0214

0.1 1.3 STANDARD NORMAL DISTRIBUTION: Table Values Represent AREA to the LEFT of the Z score. Z .00 .01 .02 .03 .04 .05 .06 .07 .08 .09 0.0 50000 50399 50798 51197 51595 $1994 .52392 52790 53188 53586 53983 54380 54776 .55172 55567 55962 .56356 .56749 57142 57535 0.2 57926 58317 58706 59095 59483 59871 .60257 .60642 .61026 .61409 0.3 .61791 .62172 .62552 .62930 .63307 .63683 .64058 .64431 .64803 .65173 0.4 .65542 .65910 .66276 .66640 .67003 .67364 67724 .68082 68439 .68793 0.5 .69146 .69497 .69847 .70194 .70540 .70884 .71226 .71566 .71904 72240 0.6 .72575 .72907 .73237 .73565 .73891 .74215 .74537 .74857 .75175 .75490 0.7 .75804 .76115 .76424 .76730 .77035 .77337 .77637 .77935 .78230 .78524 0.8 .78814 .79103 .79389 .79673 .79955 .80234 .80511 .80785 .81057 ,81327 0.9 .81594 .81859 .82121 .82381 82639 .82894 .83147 83398 .83646 .83891 1.0 .84134 .84375 .84614 .84849 .85083 .85314 .85543 .85769 .85993 .86214 1.1 .86433 .86650 .86864 .87076 .87286 .87493 .87698 .87900 .88100 .88298 1.2 .88493 .88686 .88877 .89065 .89251 .89435 .89617 .89796 .89973 .90147 .90320 .90490 90658 90824 90988 91149 .91309 91466 91621 91774 1.4 .91924 92073 92220 92364 92507 92647 92785 92922 93056 93189 1.5 .93319 193448 93574 .93699 93822 93943 94062 .94179 94295 94408 1.6 .94520 94630 94738 .94845 .94950 95053 95154 .95254 95352 .95449 1.7 95543 -95637 95728 95818 .95907 95994 .96080 .96164 96246 .96327 1.8 96407 96485 .96562 .96638 .96712 96784 .96856 .96926 .96995 97062 1.9 97128 97193 97257 97320 .97381 97441 .97500 97558 .97615 97670 2.0 97725 97778 97831 97882 97932 .97982 98030 98077 98124 98169 2.1 98257 .98300 .98341 .98382 98422 .98461 .98500 98537 .98574 2.2 .98610 98645 .98679 98713 98745 98778 .98809 .98840 98870 .98899 2.3 .98928 .98956 98983 99010 .99036 99061 .99086 .99111 99134 99158 2.4 .99180 99202 99224 .99245 99266 .99286 99305 .99324 99343 99361 2.5 .99379 .99396 .99413 .99430 99446 .99461 .99477 .99492 99506 99520 2.6 .99534 .99547 99560 99573 .99585 99598 199609 .99621 99632 99643 2.7 .99653 99664 .99674 .99683 99693 99702 199711 99720 99728 .99736 2.8 99744 99752 .99760 .99767 .99774 99781 99788 99795 .99801 .99807 2.9 99813 99819 99825 99831 99836 99841 .99846 99851 99856 .99861 3.0 99865 .99869 99874 99878 .99882 .99886 .99889 99893 99896 .99900 3.1 .99906 .99910 99913 99916 99918 .99921 99924 .99926 .99929 3.2 .99931 .99934 99936 99938 99940 99942 .99944 99946 .99948 .99950 .99952 99953 99955 .99957 .99958 99960 .99961 99962 .99964 99965 3.3