Maths Quest Maths A Year 12 for Queensland Chapter 10 The normal distribution and games of chance
WorkSHEET 10.11
WorkSHEET 10.1StatisticsName: ______
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1 / In a class test where the mean was 62% and the standard deviation 12.6%, Jan received 74%. Calculate her result as a z-score. / / 52 / On the same test, Bob received 50%. What is Bob’s result as a z-score? / / 5
3 / The heights of eight professional basketball players are (in centimetres):
188, 189, 192, 193, 194, 194, 195, 195.
(a)Calculate the mean and standard deviation of their heights (to onedecimal place).
(b)John is 190 cm tall. Express his height as a z-score compared with the basketball players’ heights (to onedecimal place). / (a)Enter the data into a calculator which provides statistical calculations.
/ 6
4 / Bill’s height, when converted to a z-score compared with the basketball players in question3, is +1. What is Bill’s height? / Bill’s height is one standard deviation above the mean height of the basketball players.
So, Bill’s height = 192.5 cm + 2.5 cm
= 195 cm / 4
5 / Ryan received 82% in his History exam where the mean was 70% and the standard deviation 10%. In his Biology exam he received 75% when the mean was 50% and the standard deviation 12%. He felt he had performed better in History than in Biology. Is this the case? Explain. /
Ryan is 1.2 standard deviations above the class mean in History and 2.1 standard deviations above the class mean in Biology. He has therefore performed better in Biology than History. / 5
6 / In international swimming the mean time for the men's 100m freestyle is 50.46 sec with a standard deviation of 0.6 sec. For the 200m freestyle, the mean time is 110.4sec with a standard deviation of 1.4 sec.
Jason’s best time for the 100 m is 48.76sec and for the 200m is 108.43sec.
If he can only enter one of these events in the competition, which one should he enter? /
Jason’s z-score for the 100 m is lower, indicating that his time is further below the mean for this event than for the 200 m event. So, he should enter the 100 m event. / 5
7 / What percentage of the scores lie:
(a)within one standard deviation of the mean?
(b)within two standard deviations of the mean?
(c)within three standard deviations of the mean? / (a)68% of the scores lie within one standard deviation either side of the mean.
(b)95% of the scores lie within two standard deviations either side of the mean.
(c)99.7% of the scores lie within three standard deviations either side of the mean. / 6
8 / On a common IQ test, the distribution is normal with a mean of 100 and a standard deviation of 12. What percentage of the scores lie between 76 and 124? /
The score of 76 is two standard deviations below the mean and the score of 124 is two standard deviations above the mean. So, 95% of the scores lie in the range 76 to 124. / 5
9 / In the above IQ test in question 8, what range of scores would account for 99.7% of the people? / 99.7% of the scores lie within three standard deviations either side of the mean.
So a score 36 below 100 to a score 36 above 100 would account for 99.7% of the people.
So the range 64 to 136 would include 99.7% of people. / 4
10 / On an end of semester maths test the mean result was 62% and the standard deviation was 12%.
What percentage of the results would lie above 86%? /
So a score of 86% lies two standard deviations above the mean. 95% of the scores lie within two standard deviations either side of the mean. This means that 5% of the scores lie outside this range. Half of these scores lie below –2 and half above +2.
Percentage of results above 86% = 2.5% / 5