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Normal Distribution: Review

Module by: Kathy Chu. E-mail the author

Based on: Normal Distribution: Review by Susan Dean, Barbara Illowsky, Ph.D.

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"} } } {}, 3≤x≤133≤x≤13 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.
Horizontal boxplot with first whisker at 0 to 2, box from 2 to 5, line at 4, and second whisker from 5 to 7.

Solution

  • a: 8
  • b: 8
  • c: P(X<k)=0.65=(k−3)∗(110)P(X<k)=0.65=(k−3)∗(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(G∣H)=P(G)P(G∣H)=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.

Solution

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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