1. Statistics for Business and Economics
Chapter 6
Inferences Based on a Single Sample: Tests of Hypothesis

2. Learning Objectives Distinguish Types of Hypotheses
Describe Hypothesis Testing Process
Explain p-Value Concept
Solve Hypothesis Testing Problems Based on a Single Sample
Explain Power of a Test
As a result of this class, you will be able to ...

3. Descriptive Statistics Inferential Statistics
Statistical Methods
Statistical Methods
Descriptive Statistics
Inferential Statistics
Hypothesis Testing
Estimation 5

4. Hypothesis Testing Concepts

5. Reject hypothesis! Not close.
Hypothesis Testing
I believe the population mean age is 50 (hypothesis).
Reject hypothesis! Not close.
Population 
Mean = 20
Random sample 

6. I believe the mean GPA of this class is 3.5!
What’s a Hypothesis?
A belief about a population parameter
Parameter is population mean, proportion, variance
Must be stated before analysis
I believe the mean GPA of this class is 3.5!
© T/Maker Co.

7. Null Hypothesis 1. Specified as H0:   Some numeric value
Written with = sign even if , or 
Example, H0:   3
2. Assumed to be true--status quo
3. Gives the sampling distribution of the data and the test statistic (null distribution)
4. Cannot be proven; only rejected in favor of the alternative.

8. Alternative Hypothesis
1. Designated Ha
2. Opposite of null hypothesis
3. Specified Ha:  < or  or > Some value
Example, Ha:  < 3
4. The research hypothesis: what you want to prove

9. Identifying Hypotheses
Example
Historically, the average amount spent in the bookstore has been $50. There is concern that it may be dropping due to the student use of the internet. Is the average amount spent in the bookstore now less than $50?
1. H0:  = 50
2. Ha:  < 50

10. Sampling Distribution
Basic Idea
Sampling Distribution
It is “impossible” get a sample mean of this value ...
... therefore, we reject the hypothesis that  = 50.
... if in fact this were the population mean 20 m
= 50
Sample Means H0

11. Basic Idea
1. If the sample mean looks as though it could have come from the sampling distribution given by the null hypothesis, then we will accept the null hypothesis.
2. If the sample mean is way out on the tail, or completely outside the sampling distribution given by the null hypothesis, we should reject the null hypothesis.
3. Only work we have to do: decide what is inside, and what is outside the distribution!
(Have to DRAW THE LINE!)

12. Where to Draw the Line
Cut point determined by (alpha-probability of error)
Typical values are .01, .05, .10
Determines “how far in” to draw the line
Outside line: rejection region
Inside line: acceptance region
Selected by researcher at start
Alpha is the “significance level of the test”

13. Rejection Region (One-Tail Test)
Sampling Distribution
Fail to reject here— test statistic in acceptance region---H0 OK
Rejection
Observed sample statistic
Region
Rejection region does NOT include critical value.
1 –  a
Acceptance
Region Ho
Sample Statistic
Critical
Value
Value

14. Rejection Regions (Two-Tailed Test)
Sampling Distribution
Rejection
Rejection
Reject H0 here!
Region
Region
Rejection region does NOT include critical value.
1 – 
1/2 a
1/2 a
Observed sample statistic
Acceptance
Region Ho
Critical
Value
Critical
Value
Value

15. Hypothesis Testing Steps9

16. Hypothesis Testing Steps
1. State H0, H1, , and n
Collect data and compute test statistic (Xbar)
Draw the line(s)
One tail “<“ alternative: a percentile of Xbar dist’n
One tail “>” alternative: 1 - a percentile
Two tail alternative: a/2 and (1 – a)/2 percentiles
4. Draw conclusions.

17. One-Tailed Z Test Thinking Challenge
You’re an analyst for Ford. Historically, Ford Focus models have averaged 32 mpg. You want to test the research hypothesis (alternative) that the average miles per gallon of the Ford Focus is now greater than 32 mpg. Similar models have a standard deviation of 4.0 mpg. You take a sample of 64 Focus models and compute a sample mean of mpg. Test at the 0.01 level of significance. Interpret the results.
Alone
Group
Class

18. One-Tailed Z Test Solution
Step: 1. 2. 3. 4.
Interpretation:

19. Using “Hypothesis Tests For Mean Calculator.jmp”
Supply the five inputs in the first five columns:
And read the results:

20. Two-Tailed Z Test Example
Does a cereal production line produce boxes containing an average of 368 grams of cereal as specified on the box---or has the value changed? To test these hypotheses, a random sample of 36 boxes showed = The company has specified  to be 12 grams. Test at the .05 level of significance. Interpret the results
368 gm.

21. Two-Tailed Z Test Solution
Step: 1. 2. 3. 4.
Interpretation:

22. Observed Significance Levels:
p-Values
Short cut: If you have these, you do not need to calculate rejection region(s) 9

23. p-Value
1. Probability of obtaining a test statistic more extreme (or than the actual sample value given H0 is true
2. Called observed level of significance
Smallest value of  H0 can be rejected
3. Used to make rejection decision
If p-value  , do not reject H0
If p-value < , reject H0

24. Hypothesis Testing Steps Using P-value (Step 3 reject regions replaced)
1. State H0, H1, , and n
Collect data and compute test statistic (Xbar)
Compute p-value
One tail “<“ alternative: P(test stat < observed val)
One tail “>” alternative: P(test stat > observed val)
Two tail alternative: Compute one-tail and double
Draw conclusions: If p < alpha, reject null
“If p is low, the null’s gotta go”

25. One-Tailed Z Test via P-value
You’re an analyst for Ford. Historically, Ford Focus models have averaged 32 mpg. You want to test the research hypothesis (alternative) that the average miles per gallon of the Ford Focus is now greater than 32 mpg. Similar models have a standard deviation of 4.0 mpg. You take a sample of 64 Focus models and compute a sample mean of mpg. Test at the 0.01 level of significance. Interpret the results.
Alone
Group
Class

26. P-value Solution
Test statistic is = 33.7 and the alternative is a “greater than” alternative. So:
p-value = P( > 33.7 given the null is true)
Easy: Just use JMP to find P( > 33.7) using the null distribution: m = 32 and s = 4/8=.5

27. Two-Tailed p-Value Solution
Just find the one-tail value and double it!

28. Two-Tailed P-value Example
You’re a Q/C inspector. You want to find out if a new machine is making electrical cords to customer specification: average breaking strength of 70 lb. with  = 3.5 lb. You take a sample of 36 cords & compute a sample mean of 69.7 lb. At the .05 level of significance, is there evidence that the machine is not meeting the average breaking strength? Determine the p-value, draw conclusions.
Use “Distribution_Calculator.jsl” to draw the p-value on the sampling distribution.
Do it the easy way using “Hypothesis Tests For Mean.jmp”

29. One Population Tests One Population Z Test t Test Mean Proportion c
(1 & 2
tail)
t Test
Mean
Proportion
Variance
(not covered) c 2
Test

30. What if  is unknown? Use the sample standard deviation, s. Then:
If the null hypothesis is true and
If the raw data are normally distributed, then
The number of number of standard deviations from Xbar to m
follows a t distribution with n-1 degrees of freedom. The p-value can be obtained using the t distribution.

31. One-Tailed t Test Example
Is the average capacity of batteries at least 140 ampere-hours? A random sample of 20 batteries had a mean of and a standard deviation of Assume a normal distribution. Test at the .05 level of significance.

32. One-Tailed t Test Solution
Step 1:
Step 2: Test Statistic:
H0:
Ha:
 =
n =
 = 140
 < 140
.05 20
Step 3:
P-value = P(t < -2.57) = (using JMP for t with 19 df)
Step 4:
Conclude: Reject H0 because p =.0093 < a = .05

33. Use Distribution_Calculator.jsl in JMP:

34. Use Hypothesis Test for Mean Calculator.jmp in JMP:

35. One-Tailed t Test Thinking Challenge
You’re a marketing analyst for Wal-Mart. Wal-Mart had teddy bears on sale last week. The weekly sales ($ 00) of bears sold in 10 stores was: At the .05 level of significance, is there evidence that the average bear sales per store is more than 5 ($ 00)? (Find the p-value where the alternative is “ > 5”)
Do “by hand” the hard way (compute t and find p)!
Do using JMP Analyze >> Distribution (directions on the next page)
Assume that the population is normally distributed.
Allow students about 10 minutes to solve this.
Note: A real problem, finally! Raw data, not summarized data!

37. Results
Step 1. Now click Teddy Bears hot spot and choose test mean
Step 2. Enter hypothesized mean
“NE” p-value
“GT” p-value
“LT” p-value

38. Z Test of Proportion 9

39. One Population Tests One Population Mean Proportion Large Small Sample
Z Test
t Test
Z Test
(1 & 2
(1 & 2
(1 & 2
tail)
tail)
tail)

40. One-Sample Z Test for Proportion
1. Assumptions
Two categorical outcomes
Population follows binomial distribution
Normal approximation can be used
does not contain 0 or 1
2. Z-test statistic for proportion
Hypothesized population proportion

41. One-Proportion Z Test Thinking Challenge
You’re an accounting manager. A year-end audit showed 4% of transactions had errors. You implement new procedures. A random sample of 500 transactions had 25 errors. Has the proportion of incorrect transactions changed at the .05 level?
Alone
Group
Class

42. One-Proportion Z Test Solution
Step 1: (H0, Ha, n, a)
Step 2: (Z test statistic)
Step 3: P-value
Step 4: Conclusion

43. Hypothesis Tests for Proportion Calculator.jmp
Using JMP Table:
Hypothesis Tests for Proportion Calculator.jmp

45. Using JMP Menus
1. Data table. Specify Analyze >> Distribution and select the variable Audit Outcome
2. Use hot spot to select : “Test Probabilities” (e.g., Test Proportions)
3. Results and p-value (Pearson)

46. Decision Making Risks

47. Errors in Making Decision
Type I Error
Reject true null hypothesis
Has serious consequences
Probability of Type I Error is (alpha)
Called level of significance
Type II Error
Do not reject false null hypothesis
Probability of Type II Error is (beta)

48. Decision Results H0: Innocent Jury Trial H0 Test Actual Situation
True
False
Accept
1 – a
Type II
Error
(b)
Reject
Type I
Error (a)
Power
(1 – b)
Actual Situation
Verdict
Innocent
Guilty
Innocent
Correct
Error
Guilty
Error
Correct

49.  &  Have an Inverse Relationship
You can’t reduce both errors simultaneously!  

50. Factors Affecting  True value of population parameter
Increases when difference with hypothesized parameter decreases
Significance level, 
Increases when decreases
Population standard deviation, 
Increases when  increases
Sample size, n
Increases when n decreases

51. Conclusion Distinguished Types of Hypotheses
Described Hypothesis Testing Process
Explained p-Value Concept
Solved Hypothesis Testing Problems Based on a Single Sample
Discussed (briefly!) Type I and Type II errors
As a result of this class, you will be able to ...