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Normal Distribution: Review

Module by: Dr. Barbara Illowsky, Susan Dean

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 1

  • 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 2

  • 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 3

  • 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 4

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(JP(J size 12{P \( J} {} and K)=0K)=0 size 12{K \) =0} {}
  • B - P(JP(J size 12{P \( J} {} or K)=0.9K)=0.9 size 12{K \) =0 "." 9} {}
  • C - P(JP(J size 12{P \( J} {} or K)=0.72K)=0.72 size 12{K \) =0 "." "72"} {}
  • D - P ( J ) ≠ P ( J ∣ K ) P ( J ) ≠ P ( J ∣ K ) size 12{P \( J \) <> P \( J \lline K \) } {}

Solution 5

  • A - False – JJ size 12{J} {} and KK size 12{K} {} are independent, so they are not mutually exclusive which would imply dependency
  • B - False
  • C - True – since P(JP(J size 12{P \( J} {} and K)≠0K)≠0 size 12{K \) <> 0} {}, then P(JP(J size 12{P \( J} {} or K)<0.09K)<0.09 size 12{K \) <0 "." "09"} {}
  • D - False – P(JP(J size 12{P \( J} {} and K)≠0K)≠0 size 12{K \) <> 0} {} are independent which implies P(J)=P(J∣K)P(J)=P(J∣K) size 12{P \( J \) =P \( J \lline K \) } {}

Exercise 6

On average, 5 students from each high school class get full scholarships to 4-year colleges. Assume that most high school classes have about 500 students.

XX = the number of students from a high school class that get full scholarships to 4-year school. Which of the following is the distribution of XX?

  • A. P(5)
  • B. B(500,5)
  • C. Exp(1/5)
  • D. N(5, (0.01)(0.99)/500)

Solution 6

A

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