Maths Made Easy Forums Standard Data & Statistics Normal Distribution

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Keymaster
Post count: 724

In a small country town, the ages of the population are normally distributed. The mean age is 38 years and the standard deviation is 12 years. What percentage of the population lies between the ages of 38-62 is closest?

$$z=\frac{38-38}{12}$$
= 0
$$z=\frac{62-38}{12}$$
= 2
since 95% is between -2 and 2, so 0 to 2 is half of that
47.5%

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Keymaster
Post count: 724

A normal distribution has mean 70 and standard deviation 5. What percentage of scores are between 65 – 70?

$$z=\frac{65-70}{5}$$

z = -1

$$z=\frac{70-70}{5}$$

z = 0

we need the percentage of scores that lie between 0 and -1<br />
If we consider between -1 and 1, it is 68%, so from -1 to 0 would be half of that.<br />
½ × 68%<br />
∴  34%

Keymaster
Post count: 724

Michael scored 80 in his English exam where the mean was 70 and the standard deviation was 10. In Maths he scored 72 and the mean was 60 and the standard deviation was 8. Michael’s mother looked at his marks and commented: “You have performed better in English than in Maths.” Is this comment accurate? Explain.

English: $$z=\frac{80-70}{10}$$
z = 1

Maths: $$z=\frac{72-60}{8}$$
z = 1.5

Since the z-score was higher for maths, his maths score was actually better compared to the class average, so the statement is not accurate.

Keymaster
Post count: 724

A wood saw cuts timber to an average length of 1200mm, with a standard deviation of 6mm. What are the limits of timber that would “very probably” lie between?

“very probably” means within 2 standard deviations

find the lower limit which means set 2 standard deviations below the mean

1200 – 2 × 6 = 1188

ind the upper limit which means set 2 standard deviations above the mean

1200 + 2 × 6 = 1212

∴ 1188mm – 1212mm

Keymaster
Post count: 724

In a normal distribution the mean is 58. A score of 70 corresponds to a standardized score of 1.5. The standard deviation of the distribution is?

$$1.5=\frac{70-58}{s}$$         substitute z = 1.5, $$\bar{x}$$= 58 and x = 70 into the formula

1.5s = 12                                     multiply both sides by sd, and calculate 70 – 58 = 12

$$s=\frac{12}{1.5}$$           multiply both sides by sd, and calculate 70 – 58 = 12

s = 8

Keymaster
Post count: 724

The mean of set of scores is 45 and the standard deviation is 5. Between what scores do 95% of the scores lie?

95% is in between z scores of -2 and 2.
So you need 2 standard deviations below and above average.
45 – 2 × 5 = 35
45 + 2 × 5 = 55
∴ 35 – 55