Confidence levels, significance levels and critical values

Lessons

• Introduction
  Using sample data, it is possible to find a Z-score relating to the claim (the test-statistic).
  
  Significance level
  
  

  Confidence Level $(1-\alpha$)	Significance Level ($\alpha$)	Critical Value ($Z_{\alpha}$)
  0.90	0.10	?
  0.95	0.05	?
  0.99	0.01	?

  
  
  Left Tail Test:
  
  .
  
  Right Tail Test:
  
  .
  
  Two tailed Test:
  
  .
  
  

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• 1.
  Finding the Critical Value
  With a significance level of $\alpha =0.075$ what is the resulting critical value of:
  
  a)

  A right-tailed test?
  
  
  b)

  A left-tailed test?
  
  
  c)

  A two-tailed test?
  
  

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• 2.
  Find the critical value from the following confidence levels for a right-tailed test:
  a)

  90% confidence
  
  
  b)

  95% confidence
  
  
  c)

  99% confidence
  
  

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• 3.
  Rejecting or Failing to Reject the Null Hypothesis
  The following Null Hypothesis and Alternative Hypothesis have been derived from a statement:
  
  

  $H_0$:

  $p \leq 0.5$
  
  

  $H_1$:

  $p$ > $0.5$
  
  Using a significance level of $0.10$ (which corresponds to a critical value of $Z_\alpha=1.28$), and a Z-score of $Z=1.40$ relating to our Null Hypothesis;
  
  Can we reject the Null Hypothesis?

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• 4.
  The following Null Hypothesis and Alternative Hypothesis have been derived from a statement:
  
  

  $H_0$:

  $\mu \geq 175 lbs$
  
  

  $H_1$:

  $\mu$ < $175 lbs$
  
  Using a 95% confidence level and a Z-score of Z=-1.50 relating to our Null Hypothesis;
  
  Can we reject the Null Hypothesis?

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• 5.
  The following claim is made.
  
  "70% of Canadians own a pet".
  
  Given that the test-statistic is Z=2.75, with a confidence level of 90% what can be said about the proportions of pet owners in Canada?

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