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Find probabilities using the normal distribution II

d) A score that is

30

points below the mean.

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2

. The Welcher Adult Intelligence Test Scale is composed of a number of subtests. On one subtest, the raw scores have a mean of

35

and a standard deviation of

6

. Assuming these raw scores form a normal distribution

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a) What number represents the

65^{\text{th}}

percentile (what number separates the lower

65\%

of the distribution)?

Given a mean, standard deviation, and a raw score, find the corresponding

2

-score. Assume the distribution is nom mean

50

, standard deviation

4, x=60

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What is the corresponding

z

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z=

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\square

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Help me solve this

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View an example

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40

5

2010

11

] A life insurance company issues a four-year unit-linked policy to a male life. The following non-unit cash flows,

NUCF_{(t)}

1

2

3

4

), are obtained at the end of each year

t

per policy in force at the start of the year

t

1

2

3

4

NUCF_{(t)}

-50

2

-43

1

-32

1

145

5

Assume that the annual mortality rate for the male life is constant at

1\%

at all ages. (i) Show that the annual internal rate of return is

6\%

. The company sets up reserves in order to zeroise future negative cash flows. The rate of interest earned on non-unit reserves is

2.5\%

pa. (ii) Calculate the net present value of the profits after zeroisation using a risk discount rate of

6\%

pa. (iii) Comment on the results obtained in (i) and (ii) above.

2

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On a popular app, users rate hair salons as

1,2,3,4

5

stars. Suppose a rating is randomly selected from all the ratings on the app. Let

X

be the number stars of the selected rating. Here is the probability distribution of

X

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\begin{tabular}{|c|c|c|c|c|c|}

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x

X

1

2

3

4

5

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P(X=x)

0

25

0

19

0

09

0

21

0

26

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For parts (a) and (b) below, find the probability that the randomly selected hair salon rating has the described number of stars.

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2

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3

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\square