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The Central Limit Theorem: Homework is part of the collection col10555 written by Barbara Illowsky and Susan Dean.

X ~ N ( 60 , 9 ) size 12{X "~" N \( "60",9 \) } {} . Suppose that you form random samples of 25 from this distribution. Let X ¯ size 12{ {overline {X}} } {} be the random variable of averages. Let ΣX size 12{ΣX} {} be the random variable of sums. For c - f , sketch the graph, shade the region, label and scale the horizontal axis for X ¯ size 12{ {overline {X}} } {} , and find the probability.

  • Sketch the distributions of X size 12{X} {} and X ¯ size 12{ {overline {X}} } {} on the same graph.
  • X ¯ size 12{ {overline {X}} "~" } {} ~
  • P ( x ¯ < 60 ) = size 12{P \( {overline {x}}<"60" \) ={}} {}
  • P ( 56 < x ¯ < 62 ) = size 12{P \( "56"<{overline {x}}<"62" \) ={}} {}
  • P ( 18 < x ¯ < 58 ) = size 12{P \( "18"<{overline {x}}<"58" \) ={}} {}
  • Σx size 12{Σ} {} ~
  • Find the minimum value for the upper quartile for the sum.
  • P ( 1400 < Σx < 1550 ) = size 12{P \( "1400"<Σx<"1550" \) ={}} {}
  • Xbar ~ N ( 60 , 9 25 ) size 12{ ital "Xbar" "~" N \( "60", { {9} over { sqrt {"25"} } } \) } {}
  • 0.5000
  • 0.8536
  • 0.1333
  • 1530.35
  • 0.8536

Determine which of the following are true and which are false. Then, in complete sentences, justify your answers.

  • When the sample size is large, the mean of X ¯ size 12{ {overline {X}} } {} is approximately equal to the mean of X size 12{X} {} .
  • When the sample size is large, X ¯ size 12{ {overline {X}} } {} is approximately normally distributed.
  • When the sample size is large, the standard deviation of X ¯ size 12{ {overline {X}} } {} is approximately the same as the standard deviation of X size 12{X} {} .

The percent of fat calories that a person in America consumes each day is normally distributed with a mean of about 36 and a standard deviation of about 10. Suppose that 16 individuals are randomly chosen.

Let X ¯ = size 12{ {overline {X}} ={}} {} average percent of fat calories.

  • X ¯ ~ size 12{ {overline {X}} "~" } {} ______ ( ______ , ______ )
  • For the group of 16, find the probability that the average percent of fat calories consumed is more than 5. Graph the situation and shade in the area to be determined.
  • Find the first quartile for the average percent of fat calories.
  • N ( 36 , 10 16 ) size 12{ ital "Xbar" "~" N \( "36", { {"10"} over { sqrt {"16"} } } \) } {}
  • 1
  • 34.31

Suppose that the distance of fly balls hit to the outfield (in baseball) is normally distributed with a mean of 250 feet and a standard deviation of 50 feet. We randomly sample 49 fly balls.

  • If X ¯ = size 12{ {overline {X}} ={}} {} average distance in feet for 49 fly balls, then X ¯ ~ size 12{ {overline {X}} "~" } {} ______ ( ______ , ______ )
  • What is the probability that the 49 balls traveled an average of less than 240 feet? Sketch the graph. Scale the horizontal axis for X ¯ size 12{ {overline {X}} } {} . Shade the region corresponding to the probability. Find the probability.
  • Find the 80th percentile of the distribution of the average of 49 fly balls.
  • N ( 250 , 50 49 ) size 12{ ital "Xbar" "~" N \( "250", { {"50"} over { sqrt {"49"} } } \) } {}
  • 0.0808
  • 256.01 feet

Suppose that the weight of open boxes of cereal in a home with children is uniformly distributed from 2 to 6 pounds. We randomly survey 64 homes with children.

  • In words, X = size 12{X={}} {}
  • X ~ size 12{X "~" } {}
  • μ X = size 12{μ rSub { size 8{X} } ={}} {}
  • σ X = size 12{σ rSub { size 8{X} } ={}} {}
  • In words, ΣX = size 12{ΣX={}} {}
  • ΣX ~ size 12{ΣX "~" } {}
  • Find the probability that the total weight of open boxes is less than 250 pounds.
  • Find the 35th percentile for the total weight of open boxes of cereal.

Suppose that the duration of a particular type of criminal trial is known to have a mean of 21 days and a standard deviation of 7 days. We randomly sample 9 trials.

  • In words, ΣX = size 12{ΣX={}} {}
  • ΣX ~ size 12{ΣX "~" } {}
  • Find the probability that the total length of the 9 trials is at least 225 days.
  • 90 percent of the total of 9 of these types of trials will last at least how long?
  • The total length of time for 9 criminal trials
  • N ( 189 , 21 ) size 12{ ital "SumX" "~" N \( 9 * "21", sqrt {9} * 7 \) } {}
  • 0.0432
  • 162.09

According to the Internal Revenue Service, the average length of time for an individual to complete (record keep, learn, prepare, copy, assemble and send) IRS Form 1040 is 10.53 hours (without any attached schedules). The distribution is unknown. Let us assume that the standard deviation is 2 hours. Suppose we randomly sample 36 taxpayers.

  • In words, X = size 12{X={}} {}
  • In words, X ¯ = size 12{ {overline {X}} ={}} {}
  • X ¯ ~ size 12{ {overline {X}} "~" } {}
  • Would you be surprised if the 36 taxpayers finished their Form 1040s in an average of more than 12 hours? Explain why or why not in complete sentences.
  • Would you be surprised if one taxpayer finished his Form 1040 in more than 12 hours? In a complete sentence, explain why.

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Source:  OpenStax, Collaborative statistics using spreadsheets. OpenStax CNX. Jan 05, 2016 Download for free at http://legacy.cnx.org/content/col11521/1.23
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