1. STATISTICAL INFERENCES
CHAPTER 4 (PART 3)
STATISTICAL INFERENCES

2. Hypothesis testing Hypothesis testing for population mean
One population mean
Two population mean
Hypothesis testing for population proportion
One population proportion
Two population proportions

3. PLEASE MEMORIZE THE STEPS!!!!!
4.3 HYPOTHESIS TESTING
Hypothesis and Test Procedures
 State the null hypothesis, and alternative hypothesis,
Determine the rejection and non-rejection regions
Calculate the value of the test statistic
Make a decision/conclusion
PLEASE MEMORIZE THE STEPS!!!!!

4. BEFORE WE BEGIN!
Inferential statistics are methods used to determine something about a population, based on the observation of a sample
Information about a population will be presented in one of two forms, as a mean (μ) or as a proportion (p)

5. Use the population mean (μ) in the hypothesis statements when the question gives you information about the population in the form of an average
e.g. “the average travel time was 40 minutes…”,
μ = 40 minutes
Use the population proportion (p) in the hypothesis statements when the question gives you information about the population in the form of a fraction, percentage, or decimal
e.g. “ 4 out of 5 dentists agree…”,
p = ⅘ or p = 80% or p = .80

6. 1. State the null hypothesis and alternative hypothesis,
Stating the Null Hypothesis is the starting point of any hypothesis testing question solution
The Null Hypothesis is the stated or assumed value of a population parameter (the mean or proportion that is being analyzed)
What the question says the population is doing
The current or reported condition

7. When trying to identify the population parameter needed for your solution, look for the following phrases:
“It is known that…”
“Previous research shows…”
“The company claims that…”
“A survey showed that…”
When writing the Null Hypothesis, make sure it includes an “=” symbol. It may look like one of the following:
e.g. H0: μ = 40 minutes
e.g. H0: μ ≤ 40 minutes
e.g. H0: μ ≥ 40 minutes

8. Alternative hypothesis
The Alternative Hypothesis accompanies the Null Hypothesis as the starting point to answering hypothesis testing questions.
When trying to identify the information needed for your Alternative Hypothesis statement, look for the following phrases:
“Is it reasonable to conclude…”
“Is there enough evidence to substantiate…”
“Does the evidence suggest…”
“Has there been a significant…”

9. There are three possible symbols to use in the Alternate Hypotheses, depending on the wording of the question
Use “≠” when the question uses words/phrases such as:
“is there a difference...?”
“is there a change...?”
Use “<” when the question uses words/phrases such as:
“is there a decrease…?”
“is there less…?”
“are there fewer…?”

10. Use “>” when the question uses words/phrases such as:
“is there a increase…?”
“is there more…?”
When writing the Alternative Hypothesis, make sure it never includes an “=” symbol. It should look similar to one of the following:
e.g. H1: μ < 40 minutes
e.g. H1: μ > 40 minutes
e.g. H1: μ ≠ 40 minutes

11. Example
A recent survey of college campuses across Perlis claims that students spend an average of 2.7 hours a day using their cell phones. A random sample of 35 UniMAP students showed an average use of 2.9 hours a day, with a standard deviation of 0.4 hours. Do UniMAP students use their cell phones more than the typical Perlis college student?

12. Step 1: Find the population information
Read the question carefully and try and find information that is being presented as, or claims to be, fact.
In the first sentence we see the phrases “A recent survey…” and “claims that…” (both are good indicators that the information we need is in that sentence)

13. Next, determine if you are working with a population average (μ) or population proportion (p)
The information is given to us in the form of an average (2.7 hours) so we know we will use μ in the Null and Alternative Hypothesis statements
So far the Null and Alternative Hypothesis statements look like this:
H0: μ hours
H1: μ hours

14. Step 2: Determine the operators (math symbols)
Read the question carefully and find the sentence that ends in “?”. It is often (but not always) the last sentence of the problem.
Examine the wording of the question sentence, looking for words/phrases that indicate which operator to use.The example question asks, “Do UniMAP students use their cell phones more than the typical Perlis college student?”

15. Because the phrase “more than” is used in the question, we will use the greater than symbol (>)
The Null and Alternate Hypothesis statements now look like this:
H0: μ hours
H1: μ > 2.7 hours
The Null and Alternate Hypothesis statements must oppose each other. So the statements now;
H0: μ ≤ hours
H1: μ > hours

16. State the null hypothesis and alternative hypothesis

17. State the null hypothesis and alternative hypothesis
2. A lecturer claim that the medical students put in more hours studying compared to other students. The mean number of hours spent studying per week for other students is 23 hours with a standard deviation of 3 hours per week.
A sample of 25 medical students was selected at random and the mean number of hours spent studying per weeks was found to be 25 hours.

18. State the null hypothesis and alternative hypothesis
3. College students claim that the cost of living off- campus is less than the cost of living on-campus. To support the claim, 36 college students who stayed off-campus were selected at random and their mean expenditure per day was RM 34 with a standard deviation of RM4. If the mean expenditure of college students staying on-campus is RM 35, can we accept the claim?

19. 2. Determine the rejection and non-rejection regions
Rejection region : reject H0
Level of significance,
Tails of a Test
Two-Tailed Test
Left-Tailed Test
Right-Tailed Test
Sign in =
<
>
Rejection Region
In both tail
In the left tail
In the right tail

20. Reject H0 if Z < - Zα or p-value < α
1. If the alternative hypothesis, H1 contains the less- than inequality symbol (<), the hypothesis is a left-tailed test.
Accept H0
Reject H0
Reject H0 if Z < - Zα or p-value < α

21. Reject H0 if Z > Zα or p-value < α
2. If the alternative hypothesis, H1 contains the greater- than inequality symbol (>), the hypothesis is a right-tailed test.
Accept H0
Reject H0
Reject H0 if Z > Zα or p-value < α

22. Reject H0 if Z < Zα/2 or Z > Zα/2
3. If the alternative hypothesis, H1 contains the not-equal-to symbol (≠), the hypothesis is a two-tailed test.
Accept H0
Reject H0
Reject H0
Reject H0 if Z < Zα/2 or Z > Zα/2
Reject H0 if p-value < α/2

23. 3. Calculate the value of test statistic
a) Population Mean, µ
is known
is unknown and

24. b) Difference between two population means,

26. c) Population proportion, p
d) Difference between two population proportions,

27. EXAMPLE 4.1

28. SOLUTION 1. H0 : µ ≤ 2400 (the claim is true)
H1 : µ > (the claim is false)
2. We reject H0 if Z > or p-value < 0.01 (right tail test)
3. Test statistic,
4. Since Z = 18.97>2.3263, we reject H0. We can conclude that the average earnings for men in managerial and professional positions are significantly higher than those for women.

29. EXAMPLE 4.2
The mean lifetime of 30 bulbs produced by Company A is 500 hours and the mean lifetime of 35 bulbs produced by Company B is 495 hours.
If the standard deviation of all bulbs produced by company A is 10 hours and the standard deviation of all bulbs produced by Company B is 15 hours, test at 1% significance level that the mean lifetime of bulbs produced by Company A is better than Company B.

30. SOLUTION H1 : µ1 - µ2 > (company A is better than company B)
1. H0 : µ1 - µ2 ≤ 0 (company A is not better than company B)
H1 : µ1 - µ2 > (company A is better than company B)
2. We reject H0 if Z > or p-value < 0.01 (right tail test)
3. Test statistic,

31. 4. Since Z = 1. 6 < 2. 3263, we failed to reject H0 (accept H0)
4. Since Z = 1.6 < , we failed to reject H0 (accept H0)./ Since p-value= > 0.01, we failed to reject H0.
We can conclude that the mean lifetime of bulbs produced by company A is not better than company B.

32. EXAMPLE 4.3
A manufacturer claimed that at least 95% of the machine equipment he supplied to a factory conformed to specifications. An examination of a sample of 200 pieces of equipment revealed that 18 were faulty. Test his claim at a significance level of 0.01.

33. SOLUTION
1. H0 : p ≥ (the claim is true/ conformed to specifications)
H1 : p < (the claim is false/ not conformed to specifications)
2. We reject H0 if Z < or p-value < 0.01 (left tail test)
3. Test statistic,
4. Since Z = < -2.33, we reject H0. We can conclude that the machine equipment supplied by the manufacturer is not conformed to specifications

34. EXAMPLE 4.4
A random sample of 200 screws manufactured by machine A and 100 screws manufactured by machine B showed 19 and 5 defective screws, respectively. Test the hypothesis that the two machines are showing different quantities of performance at 5% significance level.

35. SOLUTION H1 : p1 – p2 ≠ 0 (the machines show different quantities of
Sample: n1 = 200, x1 = 19,
n2 = 100, x2 = 5,
1. H0 : p1 – p2 = 0 (no different)
H1 : p1 – p2 ≠ 0 (the machines show different quantities of
performance)
2. We reject H0 if Z < or Z > 1.96 (two tailed test)

36. 3. Test statistic,
4. Since Z=1.35 < 1.96, we failed to reject H0 (accept H0). We can conclude that the two machines showing no different quantities of performance.