1. Chapter 7 Hypothesis Testing
7-1 Basics of Hypothesis Testing
7-2 Testing a Claim about a Mean: Large Samples
7-3 Testing a Claim about a Mean: Small Samples
7-4 Testing a Claim about a Proportion
7- 5 Testing a Claim about a Standard     Deviation (will cover with chap 8)

2. Basics of Hypothesis Testing
7-1
Basics of
Hypothesis Testing

3. Definition Hypothesis
in statistics, is a statement regarding a characteristic of one or more populations

4. Steps in Hypothesis Testing
Statement is made about the population
Evidence in collected to test the statement
Data is analyzed to assess the plausibility of the statement

5. Components of a Formal Hypothesis Test

6. Components of a Hypothesis Test
Form Hypothesis
Calculate Test Statistic
Choose Significance Level
Find Critical Value(s)
Conclusion

7. Null Hypothesis: H0
A hypothesis set up to be nullified or refuted in order to support an alternate hypothesis. When used, the null hypothesis is presumed true until statistical evidence in the form of a hypothesis test indicates otherwise. .

8. Null Hypothesis: H0
Statement about value of population parameter like m, p or s
Must contain condition of equality
=, , or 
Test the Null Hypothesis directly
Reject H0 or fail to reject H0

9. Alternative Hypothesis: H1
Must be true if H0 is false
, <, >
‘opposite’ of Null
sometimes used instead of . H1 Ha

10. Note about Forming Your Own Claims (Hypotheses)
If you are conducting a study and want to use a hypothesis test to support your claim, the claim must be worded so that it becomes the alternative hypothesis.
The null hypothesis must contain the condition of equality

11. Examples Set up the null and alternative hypothesis
The packaging on a lightbulb states that the bulb will last 500 hours. A consumer advocate would like to know if the mean lifetime of a bulb is different than 500 hours.
A drug to lower blood pressure advertises that it drops blood pressure by 20%. A doctor that prescribes this medication believes that it is less. Set up the null and alternative hypothesis. (see hw # 1)

12. Testing claims about the population proportion
Test Statistic
a value computed from the sample data that is used in making the decision about the rejection of the null hypothesis
Testing claims about the population proportion
x - µ
Z* = σ n

13. Critical Region - Set of all values of the test statistic that would cause a rejection of the null hypothesis
Critical Value - Value or values that separate the critical region from the values of the test statistics that do not lead to a rejection of the null hypothesis

14. Critical Region and Critical Value
One Tailed Test
Critical
Region
Critical Value
( z score )

15. Critical Region and Critical Value
One Tailed Test
Critical
Region
Critical Value
( z score )

16. Critical Region and Critical Value
Two Tailed Test
Critical
Regions
Critical Value
( z score )
Critical Value
( z score )

17. Significance Level Denoted by 
The probability that the test statistic will fall in the critical region when the null hypothesis is actually true.
Common choices are 0.05, 0.01, and 0.10

18. Two-tailed,Right-tailed, Left-tailed Tests
The tails in a distribution are the extreme regions bounded
by critical values.

19.  is divided equally between the two tails of the critical
Two-tailed Test
H0: µ = 100
H1: µ  100
 is divided equally between
the two tails of the critical
region
Means less than or greater than
Reject H0
Fail to reject H0
Reject H0
100
Values that differ significantly from 100

20. Right-tailed Test H0: µ  100 H1: µ > 100 Points Right Values that
Fail to reject H0
Reject H0
Values that
differ significantly
from 100
100

21. Left-tailed Test H0: µ  100 H1: µ < 100 Points Left Values that
Reject H0
Fail to reject H0
Values that
differ significantly
from 100
100

22. Conclusions in Hypothesis Testing
Traditional Method
Reject H0 if the test statistic falls in the critical region
Fail to reject H0 if the test statistic does not fall in the critical region
P-Value Method
Reject H0 if the P-value is less than or equal 
Fail to reject H0 if the P-value is greater than the 

23. P-Value Method of Testing Hypotheses
Finds the probability (P-value) of getting a result and rejects the null hypothesis if that probability is very low
Uses test statistic to find the probability.
Method used by most computer programs and calculators.
Will prefer that you use the traditional method on HW and Tests

24. Finding P-values Two tailed test One tailed test (right)
p(z>a) + p(z<-a)
One tailed test (right)
p(z>a)
One tailed test (left)
p(z<-a)
Where “a” is the value of the calculated test statistic
Used for HW # 3 – 5 – see example on next two slides

25. Just find p(z > 2.66) Determine P-value z* = 2.66 * µ = 73.4
Sample data: x = 105 or
z* = 2.66
Reject
H0: µ = 100
Fail to Reject
H0: µ = 100 *
µ = 73.4
or z = 0
z = 1.96
z* = 2.66
Just find p(z > 2.66)

26. Just find p(z > 2.66) + p(z < -2.66)
Determine P-value
Sample data: x = 105 or
z* = 2.66
Reject
H0: µ = 100
Reject
H0: µ = 100
Fail to Reject
H0: µ = 100 *
z =
µ = 73.4
or z = 0
z = 1.96
z* = 2.66
Just find p(z > 2.66) + p(z < -2.66)

27. Conclusions in Hypothesis Testing
Always test the null hypothesis
Choose one of two possible conclusions
1. Reject the H0
2. Fail to reject the H0

28. Accept versus Fail to Reject
Never “accept the null hypothesis, we will fail to reject it.
Will discuss this in more detail in a moment
We are not proving the null hypothesis
Sample evidence is not strong enough to warrant rejection (such as not enough evidence to convict a suspect – guilty vs. not guilty)
The term ‘accept’ is somewhat misleading, implying incorrectly that the null has been proven. The phrase ‘fail to reject’ represents the result more correctly.

29. Accept versus Fail to Reject

30. Conclusions in Hypothesis Testing
Need to formulate correct wording of final conclusion

31. Conclusions in Hypothesis Testing
Wording of final conclusion
1. Reject the H0
Conclusion: There is sufficient evidence to conclude……………………… (what ever H1 says)
2. Fail to reject the H0
Conclusion: There is not sufficient evidence to conclude ……………………(what ever H1 says)

32. Example State a conclusion
The proportion of college graduates how smoke is less than 27%. Reject Ho:
The mean weights of men at FLC is different from 180 lbs. Fail to Reject Ho:
Used for #6 on HW

33. Type I Error
The mistake of rejecting the null hypothesis when it is true.
(alpha) is used to represent the probability of a type I error
Example: Rejecting a claim that the mean body temperature is 98.6 degrees when the mean really does equal 98.6 (test question)

34. Type II Error
the mistake of failing to reject the null hypothesis when it is false.
ß (beta) is used to represent the probability of a type II error
Example: Failing to reject the claim that the mean body temperature is 98.6 degrees when the mean is really different from 98.6 (test question)

35. Type I and Type II Errors
True State of Nature
H0 True
H0 False
Reject H0
Correct
decision
Type I error 
Decision
Fail to Reject H0
Correct
decision
Type II error 
In this class we will focus on controlling a Type I error. However, you will have one question on the exam asking you to differentiate between the two.

36. Type I and Type II Errors
a = p(rejecting a true null hypothesis)
b = p(failing to reject a false null hypothesis)
n, a and b are all related

37. Example Identify the type I and type II error.
The mean IQ of statistics teachers is greater than 120.
Type I: We reject the mean IQ of statistics teachers is 120 when it really is 120.
Type II: We fail to reject the mean IQ of statistics teachers is 120 when it really isn’t 120.

38. Controlling Type I and Type II Errors
For any fixed sample size n , as  decreases,  increases and conversely.
To decrease both  and , increase the sample size.

39. Power of a Hypothesis Test
Definition
Power of a Hypothesis Test
is the probability (1 - ) of rejecting a false null hypothesis.
Note: No exam questions on this. Usually covered in a more advanced class in statistics.

40. Testing a claim about the mean (large samples)
7-2
Testing a claim about the mean
(large samples)

41. Traditional (or Classical) Method of Testing Hypotheses
Goal
Identify a sample result that is significantly different from the claimed value By
Comparing the test statistic to the critical value

42. Traditional (or Classical) Method of Testing Hypotheses (MAKE SURE THIS IS IN YOUR NOTES)
Determine H0 and H1. (and  if necessary)
Determine the correct test statistic and calculate.
Determine the critical values, the critical region and sketch a graph.
Determine Reject H0 or Fail to reject H0
State your conclusion in simple non technical terms.

43. Test Statistic for Testing a Claim about a Proportion
Can Use
Traditional method Or
P-value method

44. Three Methods Discussed
1) Traditional method
2) P-value method
3) Confidence intervals

45. for testing claims about population means
Assumptions
for testing claims about population means
1) The sample is a random sample.
2) The sample is large (n > 30).
a) Central limit theorem applies
b) Can use normal distribution
3) If  is unknown, we can use sample standard deviation s as estimate for .

46. Test Statistic for Claims about µ when n > 30
x - µx
Z* =  n

47. Decision Criterion
Reject the null hypothesis if the test statistic is in the critical region
Fail to reject the null hypothesis if the test statistic is not in the critical region

48. Example: A newspaper article noted that the mean life span for 35 male symphony conductors was 73.4 years, in contrast to the mean of 69.5 years for males in the general population. Test the claim that there is a difference. Assume a standard deviation of 8.7 years. Choose your own significance level.
Step 1: Set up Claim, H0, H1
Claim:  = 69.5 years
H0 :  = 69.5
H1 :   69.5
Select if necessary  level:
 = 0.05

49. Step 2: Identify the test statistic and calculate
x - µ – 69.5
z* = = = 2.65 
8.7 n 35
Care must be taken when using a calculator to find the z value. If students wish to compute the value all in one step on the calculator, parentheses will need to be placed around the numerator and the denominator.

50.  = 0.05 /2 = 0.025 (two tailed test) z = - 1.96 1.96
Step 3: Determine critical region(s) and critical value(s) & Sketch
 = 0.05
/2 = (two tailed test)
0.4750
0.4750
0.025
0.025
This is a two-tailed test because the alternative indicated ‘is not equal to’.
z =
Critical Values - Calculator

51. Step 4: Determine reject or fail to reject H0:
Sample data: x = 73.4 or
z = 2.66
Reject
H0: µ = 69.5
Reject
H0: µ = 69.5
Fail to Reject
H0: µ = 69.5 *
z =
µ = 73.4
or z = 0
z = 1.96
z = 2.66
P-value = P(z > 2.66) x 2 = .0078
REJECT H0

52. Step 5: Restate in simple nontechnical terms
Claim:  = 69.5 years
H0 :  = 69.5
H1 :   69.5
There is sufficient evident to conclude that the mean life span of symphony conductors is different from the general population. OR
There is sufficient evidence to conclude that mean life span of symphony conductors is different from 69.5 years.
REJECT

53. TI-83 Calculator Hypothesis Test using z (large sample) Press STAT
Cursor to TESTS
Choose ZTest
Choose Input: STATS
Enter σ and x and two tail, right tail or left tail
Cursor to calculate or draw
*If your input is raw data, then input your raw data in L1 then use DATA

54. Testing Claims with Confidence Intervals
We reject a claim that the population parameter has a value that is not included in the confidence interval
Typically only used for two-tailed tests
For one-tailed test the degree of confidence would be 1 – 2a (don’t worry about this)

55. Testing Claims with Confidence Intervals
Claim: mean age = 69.5 years,
where n = 35, x = 73.4 and s = 8.7
95% confidence interval of 35 conductors (that is, % of samples would contain true value µ )
70.5 < µ < 76.3
69.5 is not in this interval
Therefore it is very unlikely that µ = 69.5
Thus we reject claim µ = 69.5 (same conclusion as previously stated)

56. Testing a claim about the mean (small samples)
7- 3
Testing a claim about the mean
(small samples)

57. for testing claims about population means (student t distribution)
Assumptions
for testing claims about population means (student t distribution)
1) The sample is a random sample.
2) The sample is small (n  30).
3) The value of the population standard deviation  is unknown.
4) population is approximately normal.

58. Test Statistic for a Student t-distribution
x -µx
t* = s n
Critical Values
Found in Table A-3
Degrees of freedom (df) = n -1
Critical t values to the left of the mean are negative
Most common mistake made with this procedure is to not use Table A-3 to find the critical values. Some use Table A-2 incorrectly.

59. the population essentially
Choosing between the Normal and Student t-Distributions when Testing a Claim about a Population Mean µ
Start
Use normal distribution with
x - µx Is
n > 30 ?
Yes
Z
/ n
(If  is unknown use s instead.) No
Is the
distribution of
the population essentially
normal ? (Use a
histogram.) No
Use nonparametric methods, which don’t require a normal distribution.
Yes
Use normal distribution with
Is 
known ?
x - µx
Z
/ n No
(This case is rare.)
Use the Student t distribution
with
x - µx
t
s/ n

60. Easier Decision Tree Use z if  known or n is large Use t if
is unknown and n is small and population is approximately normal
MAKE SURE THIS IS IN YOUR NOTES

61. P-Value Method Table A-3 includes only selected values of 
Specific P-values usually cannot be found from table
Use Table to identify limits that contain the P-value – very confusing
Some calculators and computer programs will find exact P-values

62. TI-83 Calculator Hypothesis Test using t (small sample) Press STAT
Cursor to TESTS
Choose TTest
Choose Input: STATS
Enter s and x and two tail, right tail or left tail
Cursor to calculate or draw
*If your input is raw data, then input your raw data in L1 then use DATA

63. Example Sample statistics of GPA include n=20, x=2.35 and s=.7
Test the claim that the GPA is greater than 2.0
Use traditional method
Use Calculator
Find exact p-value (see excel – TDIST function)

64. Testing a claim about a proportion
7-4
Testing a claim about a proportion

65. for testing claims about population proportions
Assumptions
for testing claims about population proportions
1) The sample observations are a random sample.
2) The conditions for a binomial experiment are satisfied
If np  5 and nq  5 are satisfied we 
Use normal distribution to approximate binomial with µ = np and  = npq

66. Notation n = number of trials p = x/n (sample proportion)
p = x/n (sample proportion)
p = population proportion (used in the null hypothesis)
q = 1 - p

67. Test Statistic for Testing a Claim about a Proportion
z* =
p - p pq n

68. (determining the sample proportion of households with cable TV)
p sometimes is given directly
“10% of the observed sports cars are red”
is expressed as
p = 0.10  
p sometimes must be calculated
“96 surveyed households have cable TV
Students will need to be reminded that ‘majority’ and ‘most’ indicates more than 50% and ‘lower’ means less than.
and 54 do not” is calculated using  x
p = = = 0.64 96 n
(96+54)
(determining the sample proportion of households with cable TV)

69. CAUTION 
When the calculation of p results in a decimal with many places, store the number on your calculator and use all the decimals when evaluating the z test statistic.
Large errors can result from rounding p too much. 

70. Test Statistic for Testing a Claim about a Proportion
Z* =
p - p pq n
x np
x - µ x - np n n p - p 
z = = = = 
npq
npq pq n n

71. TI-83 Calculator Hypothesis Test using z (proportions) Press STAT
Cursor to TESTS
Choose 1-PropZTest
Enter x and n and two tail, right tail or left tail
Cursor to calculate or draw