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Salaries for teachers in a particular elementary school district are normally distributed with a mean of $44,000 and a standard deviation of $6500. We randomly survey 10 teachers from that district.

  • In words, X = size 12{X={}} {}
  • In words, X ¯ = size 12{ {overline {X}} ={}} {}
  • X ¯ ~ size 12{ {overline {X}} "~" } {}
  • In words, ΣX = size 12{ΣX={}} {}
  • ΣX ~ size 12{ΣX "~" } {}
  • Find the probability that the teachers earn a total of over $400,000.
  • Find the 90th percentile for an individual teacher’s salary.
  • Find the 90th percentile for the average teachers’ salary.
  • If we surveyed 70 teachers instead of 10, graphically, how would that change the distribution for X ¯ size 12{ {overline {X}} } {} ?
  • If each of the 70 teachers received a $3000 raise, graphically, how would that change the distribution for X ¯ size 12{ {overline {X}} } {} ?
  • N ( 44 , 000 , 6500 10 ) size 12{ ital "Xbar" "~" N \( "44","000", { {"6500"} over { sqrt {"10"} } } \) } {}
  • N ( 440,000 , ( 10 ) ( 6500 ) ) size 12{ ital "SumX" "~" N \( "10" * "44","000", sqrt { \( "10" * "6500" \) } \) } {}
  • 0.9742
  • $52,330
  • $46,634

The distribution of income in some Third World countries is considered wedge shaped (many very poor people, very few middle income people, and few to many wealthy people). Suppose we pick a country with a wedge distribution. Let the average salary be $2000 per year with a standard deviation of $8000. We randomly survey 1000 residents of that country.

  • In words, X = size 12{X={}} {}
  • In words, X ¯ = size 12{ {overline {X}} ={}} {}
  • X ¯ ~ size 12{ {overline {X}} "~" } {}
  • How is it possible for the standard deviation to be greater than the average?
  • Why is it more likely that the average of the 1000 residents will be from $2000 to $2100 than from $2100 to $2200?

The average length of a maternity stay in a U.S. hospital is said to be 2.4 days with a standard deviation of 0.9 days. We randomly survey 80 women who recently bore children in a U.S. hospital.

  • In words, X = size 12{X={}} {}
  • In words, X ¯ = size 12{ {overline {X}} ={}} {}
  • X ¯ ~ size 12{ {overline {X}} "~" } {}
  • In words, ΣX = size 12{ΣX={}} {}
  • ΣX ~ size 12{ΣX "~" } {}
  • Is it likely that an individual stayed more than 5 days in the hospital? Why or why not?
  • Is it likely that the average stay for the 80 women was more than 5 days? Why or why not?
  • Which is more likely:
    • an individual stayed more than 5 days; or
    • the average stay of 80 women was more than 5 days?
  • If we were to sum up the women’s stays, is it likely that, collectively they spent more than a year in the hospital? Why or why not?
  • N ( 2 . 4, 0 . 9 80 ) size 12{ ital "Xbar" "~" N \( 2 "." 4, { {0 "." 9} over { sqrt {"80"} } } \) } {}
  • N ( 192 , 8.05 ) size 12{ ital "SumX" "~" N \( "80" * 2 "." 4, sqrt {"80"} * 0 "." 9 \) } {}
  • Individual

In 1940 the average size of a U.S. farm was 174 acres. Let’s say that the standard deviation was 55 acres. Suppose we randomly survey 38 farmers from 1940. (Source: U.S. Dept. of Agriculture)

  • In words, X = size 12{X={}} {}
  • In words, X ¯ = size 12{ {overline {X}} ={}} {}
  • X ¯ ~ size 12{ {overline {X}} "~" } {}
  • The IQR for X ¯ size 12{ {overline {X}} } {} is from _______ acres to _______ acres.

The stock closing prices of 35 U.S. semiconductor manufacturers are given below. (Source: Wall Street Journal )

  • 8.625
  • 30.25
  • 27.625
  • 46.75
  • 32.875
  • 18.25
  • 5
  • 0.125
  • 2.9375
  • 6.875
  • 28.25
  • 24.25
  • 21
  • 1.5
  • 30.25
  • 71
  • 43.5
  • 49.25
  • 2.5625
  • 31
  • 16.5
  • 9.5
  • 18.5
  • 18
  • 9
  • 10.5
  • 16.625
  • 1.25
  • 18
  • 12.875
  • 7
  • 12.875
  • 2.875
  • 60.25
  • 29.25

  • In words, X = size 12{X={}} {}
    • x ¯ = size 12{ {overline {x}} ={}} {}
    • s x = size 12{s rSub { size 8{x} } ={}} {}
    • n = size 12{n={}} {}
  • Construct a histogram of the distribution of the averages. Start at x = 0 . 0005 size 12{x= - 0 "." "0005"} {} . Make bar widths of 10.
  • In words, describe the distribution of stock prices.
  • Randomly average 5 stock prices together. (Use a random number generator.) Continue averaging 5 pieces together until you have 10 averages. List those 10 averages.
  • Use the 10 averages from (e) to calculate:
    • x ¯ = size 12{ {overline {x}} ={}} {}
    • s x ¯ = size 12{ {overline {s rSub { size 8{x} } }} ={}} {}
  • Construct a histogram of the distribution of the averages. Start at x = 0 . 0005 size 12{x= - 0 "." "0005"} {} . Make bar widths of 10.
  • Does this histogram look like the graph in (c)?
  • In 1 - 2 complete sentences, explain why the graphs either look the same or look different?
  • Based upon the theory of the Central Limit Theorem, X ¯ ~ size 12{ {overline {X}} "~" } {}
  • $20.71; $17.31; 35
  • Exponential distribution, X ~ Exp ( 1/20.71 ) size 12{X "~" N \( "60",9 \) } {}
  • $20.71; $11.14
  • N ( 20 . 71 , 17 . 31 5 ) size 12{ ital "Xbar" "~" N \( "20" "." "71", { {"17" "." "31"} over { sqrt {5} } } \) } {}

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Source:  OpenStax, Collaborative statistics (custom lecture version modified by t. short). OpenStax CNX. Jul 15, 2013 Download for free at http://cnx.org/content/col11543/1.1
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