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Section 5

Tests of Hypothesis

Continued

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n 5 Goodness of Fit Testing

• Performing statistical analysis requires us to make

assumptions about the shape of distributions.

• We typically talk about normally distributed data.

• Goodness of Fit Testing allows us to perform a test of

hypothesis to verify these assumptions.

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n 5 Expected Value

The expected value is the probability of an event

times the sample size.

The probability of finding a defective is .01. A

sample of 50,000 will have an expected number

of defectives is

(.01)(50,000) = 500

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n 5 Developing a Theoretical Distribution

In order to accurately model and predict performance, we should statistically verify that the distribution we are using is not significantly different from the the observed data.

In order to do that we must develop a theoretical distribution.

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n 5 Dice Distribution

• Create a theoretical distribution for a pair of dice.

• How many of each outcome would you expect if

you rolled the dice 100 times?

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Sample Probability Distribution

Dice

Outcome Ways to Achieve Ways

Relative

Frequency

2

3

4

5

6

7

8

9

10

11

12

Total:

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n 5 Theoretical Normal Distribution

• For example, we may want to compare sample

observations with a theoretical normal (or

Poisson or Exponential) distribution with a given

mean and standard deviation.

• We can use the spread sheet to generate such a

distribution.

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n 5 Example

• Generate a theoretical normal distribution with a

mean of 100 and a standard deviation of 5.

Predict the number of each values you would

expect to see in a sample of 500.

• Assumption is made that we can measure to the

nearest whole number.

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n 5 Example Continued

Programming the theoretical normal distribution

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n 5 Another Example

� Show the theoretical normal distribution for 800

measurements with a mean of 60 and a standard

deviation of 4.

� Assume that the measurements are to the

nearest whole number.

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n 5 Chi Square Test

• State Hypothesis

• Identify Test Statistic

• Specify Confidence

• Calculate Test Statistic

• Identify Table Value (df = defined)

• Normal = n-3

• Poisson = n-2

• Exponential = n-2

• Binomial = n-2

• Dice = n-1

• Compare Test and Table Statistics

2

2

Frequency Expected

Frequency) Expected -Frequency Observed(∑=χ

Specified Population Sample :H

Specified Population Sample :H

1

0

≠

=

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n 5 Chi Square Values

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n 5 Example

Use the indicated data set to determine whether

or not the data are normally distributed with the

given mean and standard deviation.

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n 5 Example

A population is believed to be Poisson distributed with a mean of 6.

500 values are sampled with the distribution shown. Can it be said, with 90 percent confidence, that the sample follows the same distribution as the stated population?

0 4

1 12

2 25

3 40

4 75

5 80

6 85

7 70

8 58

9 40

10 11

Sum 500

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n 5 Example

A population is normally distributed with a mean of 90 and a standard deviation of 3.

A sample of 280 has the distribution shown on the following page. At the .05 level can we determine if the sample follows the same normal distribution?

Value Observed

Frequency

80 2

81 4

82 5

83 7

84 9

85 11

86 14

87 18

88 24

89 28

90 41

91 31

92 22

93 17

94 14

95 12

96 8

97 5

98 4

99 3

100 1

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n 5 Which Distribution?

How do we know which

distribution to try to fit?

Histogram

Process Knowledge

Trial and Error

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n 5 Practice Problems

1. Aft Tech’s corporate hotel has averaged a 78 percent occupancy rate. For the past seven days, the occupancy rate has been 83 percent. Can it be said, at the 95 percent confidence level, that occupancy has increased?

2. Ralph Zenith, noted consumer advocate, has decided to test the claims that tire manufacturer A has been making regarding manufacturer B. Ralph tested 18 of A’s tires and found an average tread life of 41,400 miles with a standard deviation of 4,250 miles. Manufacturer B’s 20 tires lasted an average 39,600 miles, with a standard deviation of 3,960 miles. At the .01 level, is there a significant difference in tire lives?

3. A large department store chain has averaged 16 complaints per day on its hotline, with a standard deviation of 4 calls per day. The average for a recent sample of 14 days was 13 complaints. At the.01 level can it be said the complaint rate has decreased?

4. One class of 20 students at State Tech had an average grade of 78 with a standard deviation of 4 in the required quality control class. Historically, students in this class had an average grade of 74. Is this class significantly different. at the .05 level, from past classes?

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n 5 Practice Problems

5. Aft Tech’s packaging inspector has discovered that on the average, plant 4 fills its packages to 96 percent of capacity, based on a sample of 20, whereas plant 6 fills its packages to only 88 percent of capacity, based on a sample of 10 packages. Can the inspector say, with 99percent confidence, that there is a significant difference between the filling percents at the two plants?

6. Is there a significant difference in the variability of the packaging levels as reported in problem 5?

7. Two samples were selected from Aft Tech’s screw machine operation. The first sample, produced on the day shift, had a mean of .87 and a standard deviation of .91, based on a sample of 12 screws. The second sample, produced on the night shift, had an average of .96 with a standard deviation of 1.09, based on a sample of 10 screws. (a) At the.05 level, is the night shift sample mean larger than the mean for the day shift? (b) At the .05 level, is he variability in the night shift different than in the day shift?

8. Aft Tech’s lawn control service has traditionally served an average of 19 customers a day. A five day sample showed Aft Tech serving anaverage of 14 customers with a standard deviation of 6.2 customers. At the .1 level, can it be said the number of customers served has decreased?

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n 5 One More Practice Problem

The instructor may provide

additional practice problems for

inferential statistics.