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8. Continuous random variables
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AustraliaAU
Year 12

8.055 Empirical rule with z-scores

Lesson
Worksheet
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Worksheet
Empirical rule with z-scores
1

If Dave scores 96 in a test that has a mean score of 128 and a standard deviation of 16, what is his z-score?

2

Frank finishes a fun run in 156 minutes. If the mean time taken to finish the race is 120 minutes and the standard deviation is 12 minutes, what is his z-score?

3

For each of the following examples, find x, the test score each student received:

a

Iain's z-score in a test is 1, the mean mark is 62\% and standard deviation is 3\%.

b

Rochelle's z-score in a test is - 3, the mean mark is 80\% and standard deviation is 4\%.

4

A general ability test has a mean score of 100 and a standard deviation of 15.

a

Paul received a score of 102 in the test, find his z-score correct to two decimal places.

b

Georgia had a z-score of 3.13, find her score in the test, correct to the nearest integer.

5

Dylan scored 90\% with a z-score of 2 in English, and 78\% with a z-score of 4 in Mathematics.

In which subject was his performance better, relative to the rest of his class?

6

Jenny scored 81\% with a z score of - 2 in English, and 72\% with a z-score of - 4 in Mathematics. In which subject was her performance better, relative to the rest of the class?

7

Ray scored 12.49 in his test, in which the mean was 7.9 and the standard deviation was 1.7.

Gwen scored 30.56 in her test, in which the mean was 20.2 and the standard deviation was 2.8.

a

Find Ray's z-score.

b

Find Gwen's z-score.

c

Which of the two had a better performance relative to the other students in their classes?

8

Marge scored 43 in her Mathematics exam, in which the mean score was 49 and the standard deviation was 5. She also scored 92.2 in her Philosophy exam, in which the mean score was 98 and the standard deviation was 2.

a

Find Marge’s z-score in Mathematics.

b

Find Marge’s z-score in Philosophy.

c

Which exam did Marge do better in relative to the rest of her class?

9

Find the area under the curve, to four decimal places, for each part of the standardised normal curves described below:

a
To the left of z = 1.45
b
To the right of z = 1.58
c
To the left of z = - 1.23
d
To the right of z = - 1.17
e
Between z = 1.52 and z = 1.87
f
Between 1.10 and 1.60 standard deviations above the mean.
10

Calculate the percentage of standardised data, to two decimal places, that is:

a
Greater than z = - 1.51
b
Between z = - 1.14 and z = 2.37
11

Calculate the probability, to four decimal places, that a z-score is:

a
Either at most - 1.08 or greater than 2.07
b
Greater than - 0.63 and at most 1.44
c
At most 1.60 given that it is greater than - 0.69
d
At most 1.03 given that it is less than 2.58
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Outcomes

ACMMM169

recognise features of the graph of the probability density function of the normal distribution with mean μ and standard deviation σ and the use of the standard normal distribution

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