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AP Statistics
Standard Normal Distributions and Standard Scores
Date _________________________
Recall that if X is N (µ, σ) then z is N (0, 1). A z score represents the number of
x
standard deviations (and direction) an observation is from its mean. z 

Find the standard score (z score) for each of the following. Draw and label the standard normal curve to
represent the score you have found.
1. SAT Verbal scores are N (500, 100)
a) X = 650
b) X = 440
c) X = 720
b) X = 132
c) X = 85
2. WISC scores for children are N (100, 15)
a) X = 140
Use the Standard Probability Table (Table A) to find the proportion of the population represented by the scores
indicated below. Draw, label, and shade the standard normal curve that corresponds to each situation.
3. SAT Verbal scores are N (500, 100)
a) X < 650
b) X > 720
c) 600 < X < 640
b) X > 80
c) 90 < X < 105
4. WISC scores for children are N (100, 15)
a) X < 140
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Use the standardized z score along with Table A to find the probabilities indicated below. Draw, label, and
shade the normal curve that corresponds to part (a). Show your work using proper probability notation.
5. Heights of females between 18 and 24 are N (65.5, 2.5). Consider females in this age group and answer each
of the following:
a) What is the probability that a randomly chosen female is shorter than 61 inches?
b) What is the probability that a randomly chosen female is less than 67.5 inches?
c) What is the probability that a randomly chosen female is taller than 59.5 inches?
d) What is the probability that a randomly chosen female is taller than 70.25 inches?
e) What is the probability that a randomly chosen female is between 50 and 60 inches tall?
f) Almost all females are between what heights?
6. Scores on a college placement test are normally distributed with a mean of 225 and a standard deviation of
30 points. Find each of the following:
a) The probability that a randomly chosen student scores above the mean
b) The probability that a randomly chosen student scores below 190
c) The probability that a randomly chosen student scores between 250 and 280
d) The probability that a randomly chosen student scores above 290
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Use z scores and the area under the standard normal curve corresponding to the percentages (or proportions)
given to find the appropriate observation (x) value. Make a sketch to represent the situation.
7. Heights of females between 18 and 24 are N (65.5, 2.5). Consider females in this age group.
a) How tall must a female be to be in the top 30% of females?
b) How tall must a female be to be in the bottom 15%?
c) How tall must a female be to be in the top half?
d) What heights correspond to the middle 50% of females?
8. Weight of elite distance runners in a study were N (63.1, 4.8) kgs.
a) If 90% of elite runners weigh less than Peter, what is Peter’s weight?
b) What would your weight have to be to put you in the lowest 5% of all runners?
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9. WISC scores for children are N (100, 15)
a) What score must a child achieve on the WISC in order to fall in the top 5%?
b) What score must a child achieve to be in the top 1%?
c) What score corresponds the lowest 2% of the population?
10. Scores for a manual dexterity test are N (25, 1.5) points.
a) Find the Q1 and the Q3 values for this distribution.
b) Would a score of 30 be considered to be an outlier for this distribution? Justify your response.
c) What is the lowest score in this distribution that would not be considered to be a low outlier?
Partial Answer Key
3. 0.9332, 0.0139, 0.0779
4. 0.9962, 0.9082, 0.3779
5. 0.0.59, 0.7781, 0.9981, 0.0287, 0.0139, (58, 73)
6. 0.5, 0.8790, 0.1696, 0.0150
7. 66.8, 62.9,  65.5 . (63.8, 67.2)
8. 69.244, 55.204
9. 124.675, 134.95, 69.25
10. 23.995 and 26.005, yes, 20.98
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