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This module provides a practice of Hypothesis Testing of Single Mean and Single Proportion as a part of Collaborative Statistics collection (col10522) by Barbara Illowsky and Susan Dean.

Student learning outcomes

  • The student will conduct a hypothesis test of a single mean with known population standard deviation.

Given

Suppose that a recent article stated that the mean time spent in jail by a first–time convicted burglar is 2.5 years. A study was then done to see if the mean time has increased in the new century. A random sample of 26 first–time convicted burglars in a recent year was picked. The mean length of time in jail from the survey was 3 years with a standard deviation of 1.8 years. Suppose that it is somehow known that the population standard deviation is 1.5. Conduct a hypothesis test to determine if the mean length of jail time has increased. The distribution of the population is normal.

Hypothesis testing: single mean

Is this a test of means or proportions?

Means

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State the null and alternative hypotheses.

  • H o size 12{H rSub { size 8{O} } } {} :
  • H a size 12{H rSub { size 8{a} } } {} :

  • H o : μ = 2 . 5 size 12{H rSub { size 8{O} } :μ=2 "." 5} {} (or, H o : μ 2 . 5 size 12{H rSub { size 8{O} } :μ<= 2 "." 5} {} )
  • H a : μ > 2 . 5 size 12{H rSub { size 8{a} } :μ>2 "." 5} {}
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Is this a right-tailed, left-tailed, or two-tailed test? How do you know?

right-tailed

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What symbol represents the Random Variable for this test?

X ¯ size 12{ {overline {X}} } {}

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In words, define the Random Variable for this test.

The mean time spent in jail for 26 first time convicted burglars

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Is the population standard deviation known and, if so, what is it?

Yes, 1.5

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Calculate the following:

  • x ¯ = size 12{ {overline {x}} ={}} {}
  • σ = size 12{σ={}} {}
  • s x = size 12{s rSub { size 8{x} } ={}} {}
  • n = size 12{n={}} {}

  • 3
  • 1.5
  • 1.8
  • 26
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Since both σ size 12{σ} {} and s x size 12{s rSub { size 8{x} } } {} are given, which should be used? In 1 -2 complete sentences, explain why.

σ size 12{σ} {}

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State the distribution to use for the hypothesis test.

X ¯ ~ N ( 2 . 5 , 1.5 26 ) size 12{ {overline {X}} "~" N \( 2 "." 5,1 "." 5 sqrt {"26"} \) } {}

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Sketch a graph of the situation. Label the horizontal axis. Mark the hypothesized mean and the sample mean x ¯ size 12{ {overline {x}} } {} . Shade the area corresponding to the p-value.

Blank horizontal axis of the sample mean.

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Find the p-value.

0.0446

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At a pre-conceived α = 0 . 05 size 12{α=0 "." "05"} {} , what is your:

  • Decision:
  • Reason for the decision:
  • Conclusion (write out in a complete sentence):

  • Reject the null hypothesis
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Discussion questions

Does it appear that the mean jail time spent for first time convicted burglars has increased? Why or why not?

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Source:  OpenStax, Collaborative statistics. OpenStax CNX. Jul 03, 2012 Download for free at http://cnx.org/content/col10522/1.40
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