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# Normal Distribution: Review

Module by: Kathy Chu. E-mail the author

The next two questions refer to: X X ~ U ( 3 , 13 ) U(3,13)

## Exercise 1

Explain which of the following are false and which are true.

• a: f(x)=110f(x)=110 size 12{f $$x$$ = { {1} over {"10"} } } {}, 3x133x13 size 12{3 <= x <= "13"} {}
• b: There is no mode.
• c: The median is less than the mean.
• d: P ( X > 10 ) = P ( X 6 ) P ( X > 10 ) = P ( X 6 ) size 12{P $$X>"10"$$ =P $$X <= 6$$ } {}

### Solution

• a: True
• b: True
• c: False – the median and the mean are the same for this symmetric distribution
• d: True

## Exercise 2

Calculate:

• a: Mean
• b: Median
• c: 65th percentile.

### Solution

• a: 8
• b: 8
• c: P(X<k)=0.65=(k3)(110)P(X<k)=0.65=(k3)(110) size 12{P $$X<k$$ =0 "." "65"= $$k - 3$$ * $${ {1} over {"10"} }$$ } {}. k=9.5k=9.5 size 12{k=9 "." 5} {}

## Exercise 3

Which of the following is true for the above box plot?

• a: 25% of the data are at most 5.
• b: There is about the same amount of data from 4 – 5 as there is from 5 – 7.
• c: There are no data values of 3.
• d: 50% of the data are 4.

### Solution

• a: False – 3434 size 12{ { {3} over {4} } } {} of the data are at most 5
• b: True – each quartile has 25% of the data
• c: False – that is unknown
• d: False – 50% of the data are 4 or less

## Exercise 4

If P(GH)=P(G)P(GH)=P(G) size 12{P $$G \lline H$$ =P $$G$$ } {}, then which of the following is correct?

• A: GG size 12{G} {} and HH size 12{H} {} are mutually exclusive events.
• B: P ( G ) = P ( H ) P ( G ) = P ( H ) size 12{P $$G$$ =P $$H$$ } {}
• C: Knowing that HH size 12{H} {} has occurred will affect the chance that GG size 12{G} {} will happen.
• D: GG size 12{G} {} and HH size 12{H} {} are independent events.

D

## Exercise 5

If P(J)=0.3P(J)=0.3 size 12{P $$J$$ =0 "." 3} {}, P(K)=0.6P(K)=0.6 size 12{P $$K$$ =0 "." 6} {}, and JJ size 12{J} {} and KK size 12{K} {} are independent events, then explain which are correct and which are incorrect.

• A: P( J and K)P( J and K) = 0 = 0
• B: P( J or K) = 0.9P( J or K) = 0.9
• C: P( J or K) = 0.72P( J or K) = 0.72
• D: P ( J ) P (J K) P ( J ) P (J K)

### Solution

• A: False - J and K are independent so they are not mutually exclusive which would imply dependency (meaning P(J and K) is not 0).
• B: False - see answer C.
• C: True - P(J or K) = P(J) + P(K) - P(J and K) = P(J) + P(K) - P(J)P(K) = 0.3 + 0.6 - (0.3)(0.6) = 0.72. Note that P(J and K) = P(J)P(K) because J and K are independent.
• D: False - J and K are independent so P(J) = P(J|K).

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