1. The z-test for the Mean of a Normal Population
We want to test, m, denote the mean of a normal population

2. Situation
Let x1, … , xn denote a sample from the normal distribution with mean m and variance s2.
Both m is unknown and s2 is known
We want to test
H0: m = m 0 (some specified value of m)
Against
HA:

3. The test statistic Acceptance Region Critical Region With this Choice
Accept H0 if:
Critical Region
Reject H0 if:
With this Choice

4. The Acceptance region:
Reject H0
Accept H0

5. The Power function of a Statistical Test
Definition: The Power Function, P(q1, …,qq) of a statistical test is defined as follows:
P(q1, …,qq) = P[ test rejects H0]
Note: if H0 is true
P(q1, …,qq) = P[rejects H0 ]
= P[Type I error] = a.
if H0 is false
P(q1, …,qq) = P[rejects H0 ]
= 1- P[Type II error] = 1 - b.

6. Graph of the Power function P(q1, …,qq)b
Power function
of ideal test a
H0 is false
H0 is false q
H0 is true

7. The Power of the z-test for the mean of a Normal population
The test statistic
The Critical Region
Reject H0 if:
With this Choice

8. The test statistic
has a Standard Normal distribution if H0 is true:
More generally it has a normal distribution with

9. hence
where
has a Standard Normal distribution
with F denoting the cdf of the a Standard Normal distribution

10. Power function for z - test

11. One and Two tailed tests

12. A statistical test Consists of : A test statistic
A Critical and Acceptance Region

13. Sometimes the Critical Region is broken into two parts and lies in both tails of the sampling distribution of the test statistic (when H0 is true).
a/2

14. Sometimes the Critical Region is lies entirely in one tail of the sampling distribution of the test statistic (when H0 is true). a z

15. When the Critical Region is broken into two parts and lies in both tails the test is called a two-tailed test
a/2

16. When the Critical Region is lies entirely in one tail, the test is called a one-tailed testz

17. The Critical Region
The set of values of the test statistic that indicate HA is true.
Whether the test is one or two tailed depends on HA
Usually
If HA is two sided than the test is two-tailed.
If HA is one sided than the test is one-tailed.

18. HA: m ≠ m0 is two sided (m < m0 or m > m0 )
HA: m < m0 is one sided
HA: m > m0 is one sided

19. Whether one uses a one or two tailed depends on the objectives of the researcher
The alternative hypothesis, HA, is the research hypothesis
Different researchers could choose different alternatives because their objectives are different

20. Example
Suppose that a beach is safe to swim if the mean level of lead in the water is 10.0 (m0) parts/million.
Water safety is going to be determined by taking n = 20 water samples and using the test statistic

21. The owner of the beach may use as his choice for H0 and HA
The owner of the beach may use as his choice for H0: m ≤ 10.0
and choice for HA: m > 10.0
and keep the beach open unless H0 is rejected
This puts the burden of proof on unsafe

22. The public Health inspector may use as his choice for H0 and HA
H0: m ≥ 10.0
and choice for HA: m < 10.0
and only allow the beach to open if H0 is rejected
This puts the burden of proof on safety

23. Descriptive significance Level
P-value
Descriptive significance Level

24. Definition
In hypothesis testing the p-value is defined to be the probability that the test statistic is as or more extreme than the observed value (assuming that the Null Hypothesis (H0) is true)

25. Examples

26. Example 1
In this example a manufacturing company a metal container that is suppose to weigh m = 42.0 Kilograms. A sample of n = 50 containers were selected and found to weigh an average of
with a standard deviation of

27. We want to test
against
Test statistic
Using a = 0.05, we would reject H0 if
z < -za/2 = or
z > za/2 = 1.960

28. Now
Since
≤ z ≤ 1.960
we would accept H0

29. H0 is accepted.
Z = 1.928
0.025
0.025
1.960
-1.960

30. the p-value is defined to be the probability that the test statistic is as or more extreme than the observed value (assuming that the Null Hypothesis (H0) is true)
Now
0.025
Z = 1.928
1.960
-1.960

31. The use of a p-value is an alternative way of reporting the results of a statistical test.
If the p-values is less than a (.05 or .01) then the null hypothesis is rejected.
If the p-values is greater than a then the null hypothesis is accepted.
The p-value gives information as to how close the null hypothesis came to being rejected or accepted.
The p-value allows the reader to use his own significance level.

32. “Students” t-test

33. The Situation
Let x1, x2, x3 , … , xn denote a sample from a normal population with mean m and standard deviation s. Both m and s are unknown.
Let
we want to test if the mean, m, is equal to some given value m0.

34. Recall: The z-test for means
The Test Statistic

35. Comments
The sampling distribution of this statistic is the standard Normal distribution
The replacement of s by s leaves this distribution unchanged only the sample size n is large.

36. For small sample sizes:
The sampling distribution of
is called “students” t distribution with n –1 degrees of freedom

37. Properties of Student’s t distribution
Similar to Standard normal distribution
Symmetric
unimodal
Centred at zero
Larger spread about zero.
The reason for this is the increased variability introduced by replacing s by s.
As the sample size increases (degrees of freedom increases) the t distribution approaches the standard normal distribution

38. t distribution
standard normal distribution

39. The Alternative Hypothesis HA
The Critical Region
ta and ta/2 are critical values under the t distribution with n – 1 degrees of freedom

40. Example
Let x1, x2, x3 , x4, x5, x6 denote weight loss from a new diet for n = 6 cases.
Assume that x1, x2, x3 , x4, x5, x6 is a sample from a normal population with mean m and standard deviation s. Both m and s are unknown.
we want to test:
New diet is not effective
versus
New diet is effective

41. The Test Statistic
The Critical region:
Reject if

42. The Data
The summary statistics:

43. The Critical Region (using a = 0.05)
The Test Statistic
The Critical Region (using a = 0.05)
Reject if
Conclusion: Accept H0:

44. The z-test for Proportions
Testing the probability of success in a binomial experiment

45. Situation A success-failure experiment has been repeated n times
The probability of success p is unknown. We want to test
H0: p = p0 (some specified value of p)
Against
HA:

46. The Data The success-failure experiment has been repeated n times
The number of successes x is observed.
Obviously if this proportion is close to p0 the Null Hypothesis should be accepted otherwise the null Hypothesis should be rejected.

47. The Test Statistic
To decide to accept or reject the Null Hypothesis (H0) we will use the test statistic
If H0 is true we should expect the test statistic z to be close to zero.
If H0 is true we should expect the test statistic z to have a standard normal distribution.
If HA is true we should expect the test statistic z to be different from zero.

48. Acceptance Region Critical Region With this Choice Accept H0 if:
Reject H0 if:
With this Choice

49. Example
In the last election the proportion of the voters who voted for the Liberal party was 0.08 (8 %)
The party is interested in determining if that percentage has changed
A sample of n = 800 voters are surveyed

50. We want to test
H0: p = 0.08 (8%)
Against
HA:

51. The Test
Decide on a = P[Type I Error] = the significance level of the test
Choose (a = 0.05)
Collect the data
The number in the sample that support the liberal party is x = 92

52. Compute the test statistic
Make the Decision
Accept H0 if:
Reject H0 if:

53. Since the test statistic is in the Critical region we decide to Reject H0
Conclude that H0: p = 0.08 (8%) is false
There is a significant difference (a = 5%) in the proportion of the voters supporting the liberal party in this election than in the last election

54. the p-value is defined to be the probability that the test statistic is as or more extreme than the observed value (assuming that the Null Hypothesis (H0) is true)
Now
0.025
Z = 1.928
1.960
-1.960