Download 3.5 Applying the Normal Distribution: Z-Scores

3.5 Applying the Normal Distribution: Z-Scores
1. Calculate the z-score for each x-value, correct to one decimal place, given the mean and
standard deviation provided.
(a) mean = 8, σ = 1.5
(i) x = 4.8
(ii) x = 9.3
(iii) x = 2.9
(b) mean = 37, σ = 5
(i) x = 27.9
(ii) x = 44.6
(iii) x = 51.5
(c) mean = 12.6, σ = 2.7
(i) x = 8.8
(ii) x = 15.1
(iii) x = 4.7
2. Explain how z-score tables are used.
3. Given a normally distributed data set whose mean is 30 and whose standard deviation is 5,
what values of x would each of the following z-scores have?
(a) z = 1.00
(b) z = 0.48
(c) z = –2.30
(d) z = –0.25
4. (a) Jean-Paul’s mark is 77 out of 100 on a test known to have a mean of 60 and a standard
deviation of 18. What percentile is he in?
(b) Sheri, who also took the test, wanted to score in the 95th percentile. She got a mark of 89
out of 100. Did she achieve her goal?
5. A 50-g package of cookies is routinely discarded if its mass has a z-score of –1.1 or less.
Research has shown that the standard deviation of masses is 2 g.
(a) What is the minimum-sized cookie that is not discarded?
(b) What percent of cookies are discarded?
6. The mean temperature in Waikiki, Hawaii is 27˚C in December with a standard deviation of
2.7˚C. The mean temperature in Key West, Florida is 24˚C in December with a standard
deviation of 4.8˚C. It is 30˚C on December 25 in both cities. Which city has a lower z-score
and what does this mean in this context?
7. For the distribution X ~ N(16, 52), calculate the corresponding z-scores for the following
x-values.
(a) x = 12.8
(b) x = 6.3
(c) x = 20.1
(d) x = 25.7
8. For the distribution N(4, 0.392), determine the percent of the data that is within the given
interval.
(a) 1.64 < x < 3.56
(b) 5 < x < 12.8
--------------------------------------------------------------------------------------------------------------------Answers
1. (a) (i) z = –2.13, (ii) z = 0.87, (iii) z = –3.4
(b) (i) z = –1.82, (ii) z = 1.52, (iii) z = 2.9
(c) (i) z = –1.41, (ii) z = 0.93, (iii) z = –2.93
2. Z-score tables tell you what proportion of normally distributed data has an equal or lesser z-score than a given
value. For example, if an element from a data sample has a z-score of –1.71, you would look up –1.7 in the
column on the left and then move across to the 0.01 column. The value given corresponds to the percent of data in
a standardized normal distribution that has a lower z-score.
3. (a) x = 35
(b) x = 32.4
(c) x = 18.5
(d) x = 28.75
4. (a) z = 0.94; the corresponding value from the z-score table is 0.8238. He is in the 82nd percentile.
(b) z = 1.61; the corresponding value from the z-score table is 0.9463. Sheri has fallen just short of her goal.
5. (a) 47.8 g
(b) 13.57%
6. Z-score for Waikiki’s temperature: z = 1.11; z-score for Key West’s temperature: z = 1.25. Waikiki has a slightly
lower z-score, which means that its temperature is closer to its mean than Key West’s temperature is.
7. (a) –0.64
(b) –1.94
(c) 0.82
(d) 1.94
8. (a) 0.13 or 13% (b) 0.0052 or .52%
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