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Two-sample Tests of Hypothesis Chapter 11
GOALS ,[object Object],[object Object],[object Object],[object Object],[object Object]
Comparing two populations – Some Examples  ,[object Object],[object Object],[object Object],[object Object],[object Object]
Comparing Two Population Means ,[object Object],[object Object],[object Object]
EXAMPLE 1 The U-Scan facility was recently installed at the Byrne Road Food-Town location. The store manager would like to know if the  mean checkout time  using the standard checkout method  is longer  than using the U-Scan. She gathered the following sample information. The time is measured from when the customer enters the line until their bags are in the cart. Hence the time includes both waiting in line and checking out.
EXAMPLE 1  continued ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
Example 1  continued ,[object Object],[object Object],[object Object]
Example 1  continued ,[object Object],The computed value of 3.13 is larger than the critical value of 2.33. Our decision is to reject the null hypothesis. The difference of .20 minutes between the mean checkout time using the standard method is too large to have occurred by chance. We conclude the U-Scan method is faster.
Two-Sample Tests about Proportions ,[object Object],[object Object],[object Object],[object Object]
Two Sample Tests of Proportions ,[object Object],[object Object]
Two Sample Tests of Proportions  continued ,[object Object]
Two Sample Tests of Proportions - Example Manelli Perfume Company recently developed a new fragrance that it plans to market under the name Heavenly. A number of market studies indicate that Heavenly has very good market potential. The Sales Department at Manelli is particularly interested in whether  there is a difference in the proportions of younger and older  women who would purchase Heavenly if it were marketed. There are two independent populations, a population consisting of the younger women and a population consisting of the older women. Each sampled woman will be asked to smell Heavenly and indicate whether she likes the fragrance well enough to purchase a bottle.
[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],Two Sample Tests of Proportions - Example
[object Object],[object Object],[object Object],Two Sample Tests of Proportions - Example
[object Object],Two Sample Tests of Proportions - Example The computed value of 2.21 is in the area of rejection.  Therefore, the null hypothesis is rejected at the .05 significance level. To put it another way, we reject the null hypothesis that the proportion of young women who would purchase Heavenly is equal to the proportion of older women who would purchase Heavenly.
Two Sample Tests of Proportions – Example (Minitab Solution)
Comparing Population Means with Unknown Population Standard Deviations (the Pooled  t -test) ,[object Object],[object Object],[object Object],[object Object]
Small sample test of means  continued ,[object Object],[object Object],[object Object]
Comparing Population Means with Unknown Population Standard Deviations (the Pooled  t -test) Owens Lawn Care, Inc., manufactures and assembles lawnmowers that are shipped to dealers throughout the United States and Canada. Two different procedures have been proposed for mounting the engine on the frame of the lawnmower. The question is: Is there a difference in the mean time to mount the engines on the frames of the lawnmowers? The first procedure was developed by longtime Owens employee Herb Welles (designated as procedure 1), and the other procedure was developed by Owens Vice President of Engineering William Atkins (designated as procedure 2). To evaluate the two methods, it was decided to conduct a time and motion study.  A sample of five employees was timed using the Welles method and six using the Atkins method. The results, in minutes, are shown on the right.  Is there a difference in the mean mounting times? Use the .10 significance level.
[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],Comparing Population Means with Unknown Population Standard Deviations (the Pooled  t -test) - Example
[object Object],[object Object],[object Object],[object Object],Comparing Population Means with Unknown Population Standard Deviations (the Pooled  t -test) - Example
[object Object],Comparing Population Means with Unknown Population Standard Deviations (the Pooled  t -test) - Example (a) Calculate the sample standard deviations
[object Object],Comparing Population Means with Unknown Population Standard Deviations (the Pooled  t -test) - Example The decision is  not to reject the null hypothesis , because  0.662 falls in the region between -1.833 and 1.833 . We conclude that there is  no difference  in the mean times to mount the engine on the frame using the two methods.  -0.662
Comparing Population Means with Unknown Population Standard Deviations (the Pooled  t -test) - Example
Comparing Population Means with Unequal Population Standard Deviations ,[object Object],[object Object],[object Object]
Comparing Population Means with Unequal Population Standard Deviations - Example ,[object Object],[object Object],[object Object],[object Object],[object Object]
Comparing Population Means with Unequal Population Standard Deviations - Example The  following dot plot provided by MINITAB shows the variances to be unequal.
[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],Comparing Population Means with Unequal Population Standard Deviations - Example
[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],Comparing Population Means with Unequal Population Standard Deviations - Example The computed value of  t  is less than the lower critical value, so our decision is to reject the null hypothesis. We conclude that the mean absorption rate for the two towels is not the same.
Minitab
Two-Sample Tests of Hypothesis: Dependent Samples ,[object Object],[object Object],[object Object],[object Object]
Hypothesis Testing Involving Paired Observations ,[object Object],Where is the mean of the differences s d   is the standard deviation of the differences n  is the number of pairs (differences)
[object Object],[object Object],Hypothesis Testing Involving Paired Observations - Example
[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],Hypothesis Testing Involving Paired Observations - Example
[object Object],[object Object],[object Object],[object Object],[object Object],Hypothesis Testing Involving Paired Observations - Example
[object Object],Hypothesis Testing Involving Paired Observations - Example The computed value of  t  is greater than the higher critical value, so our decision is to reject the null hypothesis. We conclude that  there is a difference  in the mean appraised values of the homes.
Hypothesis Testing Involving Paired Observations – Excel Example
Reasons for Pairing ,[object Object],[object Object]
End of Chapter 11

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Chapter 11

  • 1. Two-sample Tests of Hypothesis Chapter 11
  • 2.
  • 3.
  • 4.
  • 5. EXAMPLE 1 The U-Scan facility was recently installed at the Byrne Road Food-Town location. The store manager would like to know if the mean checkout time using the standard checkout method is longer than using the U-Scan. She gathered the following sample information. The time is measured from when the customer enters the line until their bags are in the cart. Hence the time includes both waiting in line and checking out.
  • 6.
  • 7.
  • 8.
  • 9.
  • 10.
  • 11.
  • 12. Two Sample Tests of Proportions - Example Manelli Perfume Company recently developed a new fragrance that it plans to market under the name Heavenly. A number of market studies indicate that Heavenly has very good market potential. The Sales Department at Manelli is particularly interested in whether there is a difference in the proportions of younger and older women who would purchase Heavenly if it were marketed. There are two independent populations, a population consisting of the younger women and a population consisting of the older women. Each sampled woman will be asked to smell Heavenly and indicate whether she likes the fragrance well enough to purchase a bottle.
  • 13.
  • 14.
  • 15.
  • 16. Two Sample Tests of Proportions – Example (Minitab Solution)
  • 17.
  • 18.
  • 19. Comparing Population Means with Unknown Population Standard Deviations (the Pooled t -test) Owens Lawn Care, Inc., manufactures and assembles lawnmowers that are shipped to dealers throughout the United States and Canada. Two different procedures have been proposed for mounting the engine on the frame of the lawnmower. The question is: Is there a difference in the mean time to mount the engines on the frames of the lawnmowers? The first procedure was developed by longtime Owens employee Herb Welles (designated as procedure 1), and the other procedure was developed by Owens Vice President of Engineering William Atkins (designated as procedure 2). To evaluate the two methods, it was decided to conduct a time and motion study. A sample of five employees was timed using the Welles method and six using the Atkins method. The results, in minutes, are shown on the right. Is there a difference in the mean mounting times? Use the .10 significance level.
  • 20.
  • 21.
  • 22.
  • 23.
  • 24. Comparing Population Means with Unknown Population Standard Deviations (the Pooled t -test) - Example
  • 25.
  • 26.
  • 27. Comparing Population Means with Unequal Population Standard Deviations - Example The following dot plot provided by MINITAB shows the variances to be unequal.
  • 28.
  • 29.
  • 31.
  • 32.
  • 33.
  • 34.
  • 35.
  • 36.
  • 37. Hypothesis Testing Involving Paired Observations – Excel Example
  • 38.