NORMAL DISTRIBUTION TEST
Multiple-choice questions (13 marks)
1 Pr(Z < 1.983), where Z is a standard normal random variable is closest to:
A 0.0237
B 0.9763
C 0.4763
D 0.0559
E 0.9441
2 If Z is a standard normal random variable, and Pr(Z > c) = 0.65, then the value of c is
closest to:
A 0.2578
B 0.2578
C 0.3853
D 0.3853
E 0.3230
3 If a random variable X is normally distributed with a mean of 100 and a standard deviation
of 10, then Pr(82 < X < 112) is closest to:
A 0.8849
B 0.0819
C 0.3037
D 0.1510
E 0.8490
4 The amount of lemonade in a 2-litre bottle is normally distributed with a mean of 2.1 litres
and a standard deviation of 0.07 litres. The proportion of bottles that actually contain more
than 2 litres is:
A 0.0540
B 0.9234
C 0.1790
D 0.0766
E 0.9460
5 If X is a normally distributed random variable with mean = 5 and standard deviation =
3, then the transformation that maps the curve of the density function of X, f(x), to the curve
of the standard normal distribution is:
x 5
A ( x, y ) ,3y
3
x 5
B ( x, y ) ,5y
3
x 3
C ( x, y ) ,5y
5
x 3
D ( x, y ) ,5y
3
y
E ( x, y ) 3( x 5),
3
6 Jolanta is a high jumper. The height she can jump is normally distributed with a mean of
1.15 m and a standard deviation of 0.05 m. On 25% of her jumps, Jolanta clears h metres
or more. What is the value of h?
A 1.12 metres
B 1.14 metres
C 1.16 metres
D 1.18 metres
E 1.20 metres
7 If X is normally distributed random variable with mean 100 and standard deviation 20,
and Z is the standard normal random variable, then the interval shaded in the diagram below
can be written as:
A Pr(0< Z <1)
B Pr(Z > 100)
C Pr(Z < 120)
D Pr(100 < Z <120)
E Pr(Z < 1)
8 The diagram shows two normal distribution curves, the scores achieved on an
assignment by a group of Year 11 students, and the scores achieved on the same
assignment by a group of Year 10 students.
Which one of the following sets of statements is true?
A The mean score for the Year 11 students is higher than the mean score for the Year 10
students, but the Year 11 marks are more variable than the Year 10 marks.
B The mean score for the Year 11 students is lower than the mean score for the Year 10
students, but the Year 11 marks are more variable than the Year 10 marks.
C The mean score for the Year 11 students is higher than the mean score for the Year 10
students, but the Year 11 marks are less variable than the Year 10 marks.
D The mean score for the Year 11 students is lower than the mean score for the Year 10
students, but the Year 11 marks are less variable than the Year 10 marks.
E The mean score for the Year 11 students is higher than the mean score for the Year 10
students, and the Year 11 marks and Year 10 marks are equally variable.
9 The marks achieved by Mikki in Mathematics, English and French, together with the
mean and the standard deviation for each subject, are given in the following table:
Standard
Subject Mark Mean ()
deviation ()
Mathematics 60 52 10
English 70 70 5
French 65 60 3
Which of the following statements is correct?
A Mikkis best subject was English, followed by French and then Mathematics.
B Mikkis best subject was Mathematics, followed by French and then English.
C Mikkis best subject was French, followed by Mathematics and then English.
D Mikkis best subject was English, followed by Mathematics and then French.
E Mikkis best subject was French, followed by English and then Mathematics.
10 Scores on an exam are known to be normally distributed, with a standard deviation of
10. If 8% of the people who sat for the exam scored more than 80 marks, then the mean
examination score is closest to:
A 72
B 62
C 57
D 66
E 50
11 A radio station has planned programming on the assumption that the mean length of songs
= 4 minutes. If it is found that 30% of songs last less than 3 minutes, then the standard
deviation of length of song is closest to:
A 1.91 minutes
B 0.96 minutes
C 1 minute
D 0.52 minutes
E 1.04 minutes
12 Suppose that X is normally distributed with a mean of 50 and a standard deviation of 6.
The value of c, such that Pr(50 c < X < 50 + c) = 0.95 is closest to:
A 11.76
B 1.96
C 1.65
D 9.87
E 2.00
13 Suppose that pulse rates of people in a certain population are normally distributed. If 70%
of people have pulse rates greater than 65 beats per minute, and 10% of people have pulse
rates of more than 80 beats per minute, then the mean and the standard deviation of pulse
rate in this population are closest to:
A = 73.3, = 8.3
B = 54.6, = 19.8
C = 69.4, = 8.3
D = 75.4, = 8.3
E = 74.4, = 14.1
Short-answer questions (10 marks)
1 For the standard normal random variable Z, if Pr(Z > a) = p, and a > find in terms of p:
a Pr(Z a)
1 mark
b Pr(a < Z < a)
1 mark
2 Let X be a normally distributed random variable with a mean = 2 and a standard
deviation = 0.5. Find:
a Pr(X < 1)
1 mark
b Pr(X > 2.5)
1 mark
3 Let X be a normally distributed random variable with a mean = 3 and a standard
deviation = 1. If Pr(Z < 1.28) = 0.90, where Z is the standard normal random variable,
find c such that Pr(X < c) = 0.90.
3 marks
4 The life of a brand of TV is approximately normally distributed with a mean life of
7 years and a standard deviation of 2.5 years. If this brand of TV is guaranteed for 2 years,
what percentage of original sales will require replacement within the guaranteed date?
3 marks
Extended-response questions (6 marks)
The gestation time (the time elapsed between conception and birth) for pregnancies without
problems in humans is approximately normally distributed with a mean of 266 days and a
standard deviation of 16 days.
a What percentage of pregnancies last between 234 and 298 days?
1 mark
b In a paternity case, a woman claims her pregnancy lasted 312 days. The supposed father of
the child had been sent away on a scientific expedition to Antarctica 312 days before the
birth. What percentage of gestation times exceed 312 days in length?
2 marks
c Suppose that 100 chosen women give birth one day at a hospital. What is the probability,
to four decimal places, that at least one of the women has a gestation period of more than
312 days in length?
3 marks
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