1. Chapter 2 Hypothesis Testing Test for one and two means
Test for one and two proportions

2. Hypothesis Tests: An Introduction
WHY WE HAVE TO DO THE HYPOTHESIS?
To make decisions about populations based on the sample information.
Example :- we wish to know whether a medicine is really effective to cure a disease. So we use a sample of patients and take their data in effect of the medicine and make decisions.
To reach the decisions, it is useful to make assumptions about the populations. Such assumptions maybe true or not and called the statistical hypothesis.

3. Hypothesis Tests: An Introduction
A statistical test of hypothesis consist of : 1. The Null hypothesis, 2. The Alternative hypothesis, 3. The test statistic 4. The rejection region & critical point 5. The conclusion

4. Definitions

5. Definitions

6. Critical Region / Rejection region:
Test Statistic:
It is a function of the sample data on which the decision is to be based.
Critical Region / Rejection region:
It is a set of values of the test statistics for which the null hypothesis will be rejected.
Critical point:
It is the first (or boundary) value in the critical region.

7. Hypothesis Testing for One Sample
One Sample Mean
One Sample Proportion

8. Procedure for hypothesis testing
1. Define the question to be tested and formulate a hypothesis for a stating the problem.
Choose the appropriate test statistic and calculate the sample statistic value. The choice of test statistics is dependent upon the probability distribution of the random variable involved in the hypothesis.
Determining the critical point and rejection region,
Compare with test statistics. whether to accept or to reject the null hypothesis (H0).
5. Draw conclusions-make a statement about our claim on population is true or false.

9. Decision to Reject
. Based on the sign of H1 (alternative hypothesis)

11. How to develop H0 & H1?

12. How to develop H0 & H1?

13. How to develop H0 & H1?

14. Example:
The average monthly earnings for women in managerial and
professional positions is RM Do men in the same positions have
average monthly earnings that are higher than those for women? A
random sample of n = 40 men in managerial and professional
positions showed and s = RM 400. Test the appropriate
hypothesis using α = 0.01.
Solution:
1) The hypothesis to be tested are:
(claim)
We use normal distribution n > 30
2) Test statistic:

15. Rejection region:
Critical point:
4) Compare:
Since > 2.33, falls in the rejection region, we reject
5) Conclusion:
Thus, we conclude that men in the same positions have average monthly earnings that are higher than those for women.

16. Example:
Aisyah makes “kerepek ubi” and sell them in packets of 100g each. 12
randomly selected packets of “kerepek ubi” are taken and their weights in g
are recorded as follows:
Perform the required hypothesis test at 5% significance level to check
whether the mean weight per packet if “kerepek ubi” is not equal to 100g.
Solution:
1) The hypothesis to be tested are:
(claim)
We use t distribution, 98
102
100 96 91 97 94
101
2) Test Statistic:

17. 3) Rejection Region: Critical point: From t-table ( ): 4) Compare: Since – < , falls in the rejection region, we reject 5) We conclude that mean weight per packet of “kerepek ubi” is not equal to 100g.
Two-tailed test

18. Exercise:
A teacher claims that the student in Class A put in more hours studying
compared to other students. The mean numbers of hours spent studying per week is 25hours with a standard deviation of 3 hours per week. A sample of 27 Class A students was selected at random and the mean number of hours spent studying per week was found to be 26hours. Can the teacher’s claim be accepted at 5% significance level?

20. Example:
A manufacturer of a detergent claimed that his detergent is at least 95%
effective in removing though stains. In a sample of 300 people who had used the
detergent, n 279 people claimed that they were satisfied with the result.
Determine whether the manufacturer’s claim is true at 1% significance level.
Solution: 3
4) Since we failed to reject
5) The manufacturer’s claim that his detergent is at least 95%
effective in removing though stains. is true at 1% significance level.

21. Exercise:
When working properly, a machine that is used to make chips for calculators
produce 4% defective chips. Whenever the machine produces more than 4%
defective chips it needs an adjustment. To check if the machine is working
properly, the quality control department at the company often takes sample
of chips and inspects them to determine if they are good or defective. One
such random sample of 200 chips taken recently from the production line
contained 14 defective chips. Test at the 5% significance level whether or not
the machine needs an adjustment.

22. Hypothesis Testing for Two Sample
Difference between Two Sample Mean
Difference between Two Sample Proportion

23. Hypothesis Testing for the Differences between Two Population Mean,
Test hypothesis:
Alternative hypothesis
Rejection Region

24. Test statistics:

25. For two small and independent samples
taken from two normally distributed
populations.

26. Example:
The mean lifetime of 30 batteries produced by company A is 50 hours and the
mean lifetime of 35 bulbs produced by company B is 48 hours. If the standard
deviation of all bulbs produced by company A is 3 hour and the standard
deviation of all bulbs produced by company B is 3.5 hours. Test at 1 %
significance level that the mean lifetime of bulbs produced by Company A is
better than that of company B (claim).
Solution:
4) We reject The mean lifetime of bulbs produced by company A is better than that of company B at 1% significance level.

27. Example:
A mathematic placement test was given to two classes of 45 and 55 student
respectively . In the first class the mean grade was 75 with a standard
deviation of 8, while in the second class the mean grade was 80 with a
standard deviation of 7. Is there a significant difference between the
performances of the two classes at 5% level of significance? Assume the
population variances are equal /
Solution:

28. 3) 4) Since so we reject So there is a significant difference between the performance of the two classes at 5% level of significance.

29. Exercise: A sample of 60 maids from country A earn an average of RM300 per week with a standard deviation of RM16, while a sample of 60 maids from country B earn an average of RM250 per week with a standard deviation of RM18. Test at 5% significance level that country A maids average earning exceed country B maids average earning more than RM40 per week (claim).

31. Rejection Region:

32. Example: In a process to reduce the number of death due the dengue fever, two district, district A and district B each consists of 150 people who have developed symptoms of the fever were taken as samples. The people in district A is given a new medication in addition to the usual ones but the people in district B is given only the usual medication. It was found that, from district A and from district B, 120 and 90 people respectively recover from the fever. Test the hypothesis that the new medication helps to cure the fever using a level of significance of 5% (claim). Solution:

33. 3) 4) Since , so we reject . The new medication helps to cure the fever at 5% significance level.

34. Exercise: A researcher wanted to estimate the difference between the percentages of two toothpaste users who will never switch to other toothpaste. In a sample of 500 users of toothpaste A taken by the researcher, 100 said that they will never switched to another toothpaste. In another sample of 400 users of toothpaste B taken by the same researcher, 68 said that they will never switched to other toothpaste. At the significance level 1%, can we conclude that the proportion of users of toothpaste A who will never switch to other toothpaste is higher than the proportion of users of toothpaste B who will never switch to other toothpaste?