Tests of Hypotheses about the mean continued

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Tests of Hypothesesabout the mean - continued
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Reminder

• Two types of hypotheses
• H0 - the null hypothesis (e.g. µ24)
• H1 - the alternative hypothesis (e.g. µgt24)
• Test statistic
• P-value probability of obtaining values as
  extreme as or more extreme than the test
  statistic
• e.g., P(Z-2)0.0228
• Decision at the a significance level
• Reject H0 if p-valuelta

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Testing hypotheses using a confidence interval

• Example
• A certain maintenance medication is supposed to
  contain a mean of 245 ppm of a particular
  chemical. If the concentration is too low, the
  medication may not be effective if it is too
  high, there may be serious side effects. The
  manufacturer takes a random sample of 25 portions
  and finds the mean to be 247 ppm. Assume
  concentrations to be normal with a standard
  deviation of 5 ppm. Is there evidence that
  concentrations differ significantly (a5) from
  the target level of 245 ppm?
• Hypotheses
• H0 µ245
• H1 µ?245

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First, lets examine the Z test statistic

• Test statistic
• P-value
• 2P(Zgt2)2(0.0228)0.0456
• Decision at 5 significance level
• P-valuegta ? reject H0
• The concentration differs from 245

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Now, examine the hypotheses using a confidence
interval

• a 0.05 ? confidence level is 1- a 95
• 95 CI
• 245.04 , 248.96
• We are 95 certain that the mean concentration
  is between 245.04 and 248.96.
• Since 245 is outside this CI - reject H0.
• The concentration differs from 245

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Examine the hypotheses using a confidence interval

• H0 µµ0
• H1 µ?µ0
• If µ0 is outside the confidence interval, then
  we reject the null hypothesis at the a
  significance level.
• Note this method is good for testing two-sided
  hypotheses only
• confidence interval

µ0
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Example

• Suppose a claim is made that the mean weight µ
  for a population of male runners is 57.5 kg. A
  random sample of size 24 yields . s
  is known to be 5 kg.
• Based on this, test the following hypotheses
• H0 µ57.5
• H1 µ?57.5
• Answer using
• a) A Z test statistic
• b) A confidence interval

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• a)
• Test statistic
• P-value
• 2P(Zgt2.45)2(1-.9929)2(.0071).0142
• Decision at 5 significance level
• P-valuelta ? reject H0
• Conclusion
• Mean weight differs from 57.5

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• b)
• a 0.05 ? confidence level is 1- a 95
• 95 CI
• 58 , 62
• 57.5 is outside this CI - reject H0.
• Mean weight differs from 57.5
• question? Would you reject H0 µ59 vs. H1
  µ?59?
• No, because 59 is in the interval 58, 62

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Testing hypotheses using Minitab

• In a certain university, the average grade in
  statistics courses is 80, and s11.
• A teacher at that university wanted to examine
  whether her students received higher grades than
  the rest of the stat classes. She took a sample
  of 30 students and recorded their grades
• hypotheses
• H0µ80
• H1µgt80
• data are
• mean

95 100 82 76 75 83 75 96 75 98 79 80 79 75 100 91
81 78 100 72 94 80 87 100 97 91 70 89 99 54
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• Test statistic
• P-value
• P(Zgt2.51)1-0.99400.006
• Decision at 5 significance level
• P-valuelta ? reject H0
• conclusion
• The grades are higher than 80

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Minitab

• .\test of hypotheses.MPJ

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Column of scores
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Choose Stat gt Basic Statistics gt1-Sample Z
Pick options
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In the options window pick the alternative
hypothesis
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In the session window
One-Sample Z scores Test of mu 80 vs mu gt
80 The assumed sigma 11 Variable N
Mean StDev SE Mean scores 30
85.03 11.51 2.01 Variable 95.0
Lower Bound Z P scores
81.73 2.51 0.006
Test statistic Z 2.51 P-value
0.006 Decision reject H0 at the 5 significance
level conclusion The grades are higher than 80
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Use Minitab to build a confidence interval
Choose Stat gt Basic Statistics gt1-Sample Z
Pick options
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In the options window pick the two-sided
alternative hypothesis
Pick not equal Because a confidence interval is
like a two sided hypothesis
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In the session window
One-Sample Z scores Test of mu 80 vs mu not
80 The assumed sigma 11 Variable N
Mean StDev SE Mean scores 30
85.03 11.51 2.01 Variable
95.0 CI Z P scores
( 81.10, 88.97) 2.51 0.012
A 95 CI 81.10, 88.97 Equivalent to testing
hypotheses H0µ80 H1µ?80
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Questions

• 1. Suppose H0 was rejected at a.05. Answer
  the following questions as Yes, No, Cannot
  tell
• Would H0 also be rejected at a.03?
• Would H0 also be rejected at a.08?
• Is the p-value larger than .05?

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• 2. Suppose H0 was not rejected at a.05.
  Answer the following questions as Yes, No,
  Cannot tell
• Would H0 be rejected at a.03?
• Would H0 be rejected at a.08?
• Is the p-value larger than .05?

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• 3. A 95 confidence interval for the mean time
  (in hours) to complete an audit task is
  7.04,7.76.
• Use the relation between confidence intervals
  and two-sided tests to examine the following sets
  of hypotheses
• (a) H0 µ7.5 H1 µ?7.5 (a.05)
• (b) H0 µ7 H1 µ?7 (a.05)

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• 4. A 90 confidence interval for the mean is
  20.1, 23.5.
• We can use the relation between confidence
  intervals and two-sided tests to examine
  hypotheses about the mean
• At what level of significance, a, can we test
  these hypotheses based on the confidence
  interval?
• a.01
• a.025
• a.05
• a.1

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• 5. A 90 confidence interval for the mean is
  20.1, 23.5 has been used for testing the
  following hypotheses
• H0 µ19 H1 µ? 19
• H0 is rejected at 10 significance level
• (19 is outside the CI)
• At what level of significance, a, can we still
  reject H0 µ19?
• Answer
• We can reject H0 for alt0.1