# Hypothesis Testing for the Mean and Variance of a Population Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College.

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Hypothesis Testing for the Mean and Variance of a Population Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

Hypothesis Testing on the Mean Null hypothesis (H o ): A statement concerning a population parameter. Alternative hypothesis: A statement in contradiction of the null hypothesis Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

Type I and Type II Errors Actual Situation H o True | H o False Conclusion Fail to Reject H o Reject H o Correct decision Type I error Type II error  = probability of rejecting the H o when H o is true (Type I error)  = probability of failing to rejecting the H o when H o is false (Type II error) Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

Hypothesis Testing Process Determine the H o and H a Determine the significance level Compare the sample mean (variance) to the hypothesized mean (variance). Decide whether to Fail to reject or Reject H o Determine what the decision means in reference to the problem. Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

Example H o :  = 5.9 H a :   5.9  =.05 = P(rejecting H o when H o is true) critical value = ± 1.96 Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

Decision Rule Figure 8.1 Figure 8.2 Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

Because - 2.53 < - 1.96, we reject H o thus we conclude that the average population male height is not equal to 5.9 Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

Hypothesis Testing 5 Step Procedure Set up the null and alternative hypothesis Define the test statistic Define the rejection region Calculate the test statistic Give a conclusion in terms of the problem Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

Example 8.2 Define the hypotheses H o :  = 400 H a :   400 Define the test statistic Define the rejection region (fig8.5) –reject H o if Z > 1.645 or Z < -1.645 Fig 8.5 Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

Calculate the value of the test statistic State the conclusion –There is sufficient evidence to conclude that the average lifetime of Everglo bulbs is not 400 hours Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

Power of a Statistical Test  = P(fail to reject H o when H o is false) 1-  = P(rejecting H o when H o is false) 1-  = the Power of the test Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

One Tailed Test for  Example 8.4 H o :   32.5 H a :  < 32.5 Fig 8.8 reject H 0 if Z  X  32.5 s / n  1.645 Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

Results support the claim that the average mileage for the Bullet is less than 32.5 mpg. Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

p-Value The p-value is the value of  at which the hypothesis test procedure changes conclusions based on a given set of data. It is the largest value of  for which you will fail to reject H o Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

Procedure for Finding the p-Value For H a :    o p = 2 * (area outside Z * ) For H a :  >  o p = area to the right of Z * For H a :  <  o p = area to the left of Z * Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

Fir 8.14 Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

Interpreting the p-Value Classical Approach –reject H o if p-value <  –fail to reject H o is p-value   General rule of thumb –reject H o if p-value is small (p <.01) –fail to reject H o is p-value is large (p >.1) Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

Hypothesis Testing on the Mean of a Normal Population: Small Sample Normal population Population standard deviation unknown Small sample Student t distribution Nonparametric procedure Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

Example 8.8 H o :  = 10 H a :   10 reject H o if |t | > t.025,17 = 2.11 Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

Because 1.83 < 2.11 we fail to reject H o Insufficient evidence to reject the H o that the average output voltage is different from 10 volts Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

Small-Sample Tests on a Normal Population Mean Two-tailed test H o :  =  o H a :    o reject H o if |t*| > t  /2, n-1 One tail test H o :  =  O H a :  >  O H a :  <  O reject H o if t* > t , n- 1 reject H o if t* < t , n-1 Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

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