Area under the normal distribution and z-scores

Question 1

How do you obtain a z score?

Answer 1

By subtracting the population mean from x
and then dividing by the population standard deviation

Question 2

What is the z score formula?

Answer 2

z = (x - population mean) / population SD

Question 3

What does the z score measure?

Answer 3

How far away your sample is from the population mean in multiples of the standard deviation

Question 4

A human sampled at random from the population has an IQ of 120
what is the associated z-score?

Human mean = 100
Human SD = 15

Answer 4

z = (x - pop. mean) / pop. SD
z = (120-100) / 15 = 1.33

Answer = 1.33

Question 5

A ziltoidian sampled at random from the population has an IQ of
120 what is the associated z-score?

Ziltoidian mean = 100
Ziltoidian SD = 10

Answer 5

z = (x - pop. mean) / pop. SD
z = (120 - 100) / 10 = 2

Question 6

If the z-score is x SD below the mean…

Answer 6

The result is worse than the population average

Question 7

If the z-score is x SD above the mean…

Answer 7

The result is better than the population average

Question 8

If the z-score is high…?

Answer 8

This means that the data point is many standard deviations away from the mean (there’s a low probability of data above this z -score

Question 9

Assume that UK adult height in cm follows a normal distribution with population mean μ = 170cm and population s.d. σ = 14cm.

Your friend (an adult from the UK) has a height of 212cm. What is the associated z-score for your friend’s height?

a.	42

b.	3.00

c.	-3.00

d.	199.86

Answer 9

z = (x - pop. mean) / pop. SD
z = (212 - 170) / 14
z = 3.00

Answer = B

Question 10

Assume that UK adult height in cm follows a normal distribution with population mean μ = 170cm and population s.d. σ = 14cm.

Your friend (an adult from the UK) has a height of 212cm. How may standard deviations above the mean is your friend’s height?

a.	1

b.	2

c.	-3

d.	3

Answer 10

z = (x - pop. mean) / pop. SD
z = (212 - 170) / 14
z = 3.00

Answer = D

Question 11

A population of visiting aliens from the planet Ziltoidia 10 have highly developed intellectual abilities.They produce IQ scores that are normally distributed but with higher mean (μ = 150) and lower s.d. (σ = 5) than humans. You befriend a Ziltoidian called Zlarg with IQ of 145.

i) What is the z-score associated with Zlarg’s IQ?

ii) Is Zlarg smart when compared to other Ziltoidians?

iii) If Zlarg were actually a human, an IQ of 145 would be associated with what z-score?

a. i) -1
ii) Yes - an IQ of 145 is 3 s.d. higher than that of the average human so Zlarg is very smart
iii) 3

b. i) -1
ii) No - an IQ of 145 is 1 s.d. lower than that of the average Ziltoidian so Zlarg is not very smart
iii) 3

c. i) 3
ii) No - an IQ of 145 is 1 s.d. lower than that of the average Ziltoidian so Zlarg is not very smart
iii) -1

d. i) 3
ii) Yes - an IQ of 145 is 3 s.d. higher than that of the average human so Zlarg is very smart
iii) -1

Answer 11

z = (x - pop. mean) / pop. SD
z = (145 - 150) / 5

i) z = -1
ii) IQ of 145 is below average so Zlarg is not very smart
iii) 3

Answer = B