Lecture 15- Normal

Question 1

What is a normal distribution?/ What are its features?

Answer 1

-The normal distribution is one of the most widely used and recognized
probability distributions used for continuous random variables.
-It has a symmetric “bell shape” (line down center is like a mirror)
-The mean can take on any value (positive or negative) while the variance has to be positive

Question 2

What is a standard normal distribution?

Answer 2

The standard normal (Z) is a special case of the normal distribution with:

• µ = 0 (mean)
• σ^2 = 1 (variance)

Question 3

What does increasing the mean do for a normal distribution?

Answer 3

Shifts the curve put doesn’t change it’s shape

Question 4

What does increasing the variance do for a normal distribution curve?

Answer 4

Spreads the curve (widens it) but does not alter the center

Question 5

What is a probability density function? How does this compare to a histogram?

Answer 5

• Relative frequency histogram represents a sample (smaller number of
  individuals) .
• Probability density function represents a population (large number of
  individuals) .

Question 6

What does the area under a normal distribution curve represent?

Answer 6

Probabilities (it’s not the height of the curve that’s important!)

Question 7

What does the total area under a normal distribution curve equal?

Answer 7

1

Question 8

What function in R gives the area under a curve?

Answer 8

pnorm (q=, mean=, sd=, lower.tail=TRUE)

• q is quantile so you enter the value you want less than (less than or equal doesn’t matter in this case)
• mean and sd will default if you leave them blank to a standard normal (Z) so mean=0, sd=1 (will need to manually enter in values if distribution isn’t standard)

Question 9

Find Pr(0 < Z < 1.64)…

area under curve of normal distribution

Answer 9

pnorm(1.64) - pnorm(0)

Probability = 0.4495 4DP

Question 10

Find Pr(Z > 1.64)…

Answer 10

1 - pnorm(1.64) (this is the way to do it if you always keep lower tail= TRUE)
Probability = 0.0505 4DP

Question 11

Find Pr(1 < Z < 1.64)

Answer 11

pnorm(1.64) - pnorm(1)

Probability = 0.1082 4DP

Question 12

Find Pr(−1 < Z < 1.64)

Answer 12

pnorm(1.64) - pnorm(-1)
Probability = 0.7908 4DP

Be careful with the negative sign and brackets- r formats it weird

Question 13

Find the value Z above which 25% of the area lies.

Answer 13

qnorm(0.75)

Z = 0.6745 4DP

Question 14

How are normal quantiles found i.e. working in reverse?

Answer 14

qnorm(p, mean = 0, sd = 1, lower.tail = TRUE)

Question 15

Assume that heights of students enrolled in 100 level university papers
have a normal distribution with 
mean µX = 170cm 
variance σ2X =100
X ∼ N(170, 100)

1) Find the proportion of students with a height between 180-190cm…
2) Find the percentage of students taller than 185cm…
3) Find the height which is exceeded by 10% of students…

Answer 15

1) pnorm(190,mean = 170, sd=10)-pnorm(180,mean = 170,sd=10)
Probability = 0.1359

2) pnorm(185,mean = 170,sd=10,lower.tail = FALSE) or
1-pnorm(185,mean = 170,sd=10)
Probability = 0.0668 4DP
(Percentage = 6.68%)

3)qnorm(0.1,mean = 170,sd=10,lower.tail = FALSE) or
qnorm(0.9,mean = 170,sd=10)
Height = 182.82 cm

Question 16

What is the method used for putting any normal distribution onto a standardized scale?

Answer 16

Calculating a Z score
This is done by…
(score (X) -mean)/ standard deviation

Question 17

How should you interpret the default for qnorm: lower.tail= TRUE?

Answer 17

What is the value of the normal distribution defined with the mean and sd below which an are of exactly p occurs