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10.12 Practice 1: single mean, known population standard deviation

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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

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} {}

Is this a right-tailed, left-tailed, or two-tailed test? How do you know?

right-tailed

What symbol represents the Random Variable for this test?

X ¯ size 12{ {overline {X}} } {}

In words, define the Random Variable for this test.

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

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

Yes, 1.5

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

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} {}

State the distribution to use for the hypothesis test.

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

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.

Find the p-value.

0.0446

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

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, Elementary statistics. OpenStax CNX. Dec 30, 2013 Download for free at http://cnx.org/content/col10966/1.4
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