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8.03 Introducing the normal distribution

Worksheet
Normal distribution
1

State whether the following statements are true of the normal distribution curve:

a

There are the same number of scores above and below the mean.

b

The curve is symmetrical.

c

The fewest scores lie around the mean.

d

The curve is asymmetrical.

e

The spread of the normal distribution changes depending on the mean.

f

A higher mean will result in a skewed curve.

g

The mean, median and mode all have the same value.

h

The spread of the normal distribution changes depending on the standard deviation.

2

State whether the following graphs of data are normally distributed:

a
b
c
d
e
f
g
h
i
3

State whether the following histograms are normally distributed:

a
b
c
4

Given a normal distribution, state the percentage of people that lie:

a

Above the mean.

b

Below the mean.

5

Consider the given normal distribution:

a

State the mean.

b

State the median.

c

State the mode.

6

The mean of a set of scores, denoted by \mu is 51 and the standard deviation, denoted by \sigma is 16. Find the value of the following:

a

\mu - \sigma

b

\mu + 2 \sigma

c

\mu + 3 \sigma

d

\mu - 2 \sigma

e

\mu + 0.5 \sigma

f

\mu - \dfrac{2 \sigma}{3}

7

A data set has a mean of 80 and a standard deviation of 3. Find the score that is 3 standard deviations above the mean.

8

The following graph shows the distribution of females’ heights in a population:

a

If 34\% of females lie between 157\text{ cm} and 163\text{ cm}, find the percentage of females that lie between 163\text{ cm} and 169\text{ cm}.

b

State the value of 1 standard deviation.

9

The results from an exam were approximately normally distributed. The mean score was 65, with a standard deviation of 9. The points marked on the horizontal axis are separated by 1 standard deviation:

a

State the value of x on the graph.

b

State the value of y on the graph.

10

Assume the mass of sumo wrestlers is normally distributed, with a mean mass of 158\text{ kg} and a standard deviation of 10\text{ kg}.

a

State the percentage of wrestler that would weigh more than 158\text{ kg}.

b

How far below the mean is a sumo wrestler who weighs 148\text{ kg}.

11

In a male population, the mean height was 175\text{ cm}, with a standard deviation of 7\text{ cm}.

a

State the height of Bob who is 4 standard deviations above the mean.

b

State the height of John who is 2 standard deviations below the mean.

12

A sample of professional basketball players is normally distributed and gives the mean height as 199\text{ cm} with a standard deviation of 10\text{ cm}. State the height of a basketball player who is:

a

3 standard deviations above the mean

b

1.5 standard deviations below the mean

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MA12-8

solves problems using appropriate statistical processes

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