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1 CHAPTER 4 (PART 3) STATISTICAL INFERENCES

1 CHAPTER 4 (PART 3) STATISTICAL INFERENCES. 2 Hypothes is testing Hypothesis testing for population mean One population mean Two population mean Hypothesis

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Page 1: 1 CHAPTER 4 (PART 3) STATISTICAL INFERENCES. 2 Hypothes is testing Hypothesis testing for population mean One population mean Two population mean Hypothesis

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CHAPTER 4 (PART 3)

STATISTICAL INFERENCES

Page 2: 1 CHAPTER 4 (PART 3) STATISTICAL INFERENCES. 2 Hypothes is testing Hypothesis testing for population mean One population mean Two population mean Hypothesis

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Page 3: 1 CHAPTER 4 (PART 3) STATISTICAL INFERENCES. 2 Hypothes is testing Hypothesis testing for population mean One population mean Two population mean Hypothesis

Hypothesis and Test Procedures

1. State the null hypothesis, and alternative hypothesis,

2.Determine the rejection and non-rejection regions

3.Calculate the value of the test statistic

4.Make a decision/conclusion

3

0H

1H

4.3 HYPOTHESIS TESTING4.3 HYPOTHESIS TESTING

PLEASE MEMORIZE THE STEPS!!!!!

Page 4: 1 CHAPTER 4 (PART 3) STATISTICAL INFERENCES. 2 Hypothes is testing Hypothesis testing for population mean One population mean Two population mean Hypothesis

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)

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• 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

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Page 6: 1 CHAPTER 4 (PART 3) STATISTICAL INFERENCES. 2 Hypothes is testing Hypothesis testing for population mean One population mean Two population mean Hypothesis

1. State the null hypothesis and alternative hypothesis,

Null 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

STEPS FOR HYPOTHESIS TESTING

0H

Page 7: 1 CHAPTER 4 (PART 3) STATISTICAL INFERENCES. 2 Hypothes is testing Hypothesis testing for population mean One population mean Two population mean Hypothesis

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

STEPS FOR HYPOTHESIS TESTING

Page 8: 1 CHAPTER 4 (PART 3) STATISTICAL INFERENCES. 2 Hypothes is testing Hypothesis testing for population mean One population mean Two population mean Hypothesis

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…”

STEPS FOR HYPOTHESIS TESTING

Page 9: 1 CHAPTER 4 (PART 3) STATISTICAL INFERENCES. 2 Hypothes is testing Hypothesis testing for population mean One population mean Two population mean Hypothesis

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…?”

STEPS FOR HYPOTHESIS TESTING

Page 10: 1 CHAPTER 4 (PART 3) STATISTICAL INFERENCES. 2 Hypothes is testing Hypothesis testing for population mean One population mean Two population mean Hypothesis

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

STEPS FOR HYPOTHESIS TESTING

Page 11: 1 CHAPTER 4 (PART 3) STATISTICAL INFERENCES. 2 Hypothes is testing Hypothesis testing for population mean One population mean Two population mean Hypothesis

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?

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Page 12: 1 CHAPTER 4 (PART 3) STATISTICAL INFERENCES. 2 Hypothes is testing Hypothesis testing for population mean One population mean Two population mean Hypothesis

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)

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• 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 2.7 hours

H1: μ 2.7 hours13

Page 14: 1 CHAPTER 4 (PART 3) STATISTICAL INFERENCES. 2 Hypothes is testing Hypothesis testing for population mean One population mean Two population mean Hypothesis

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?”

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Page 15: 1 CHAPTER 4 (PART 3) STATISTICAL INFERENCES. 2 Hypothes is testing Hypothesis testing for population mean One population mean Two population mean Hypothesis

• 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 2.7 hours

H1: μ > 2.7 hours

• The Null and Alternate Hypothesis statements must oppose each other. So the statements now;

H0: μ ≤ 2.7 hours

H1: μ > 2.7 hours15

Page 16: 1 CHAPTER 4 (PART 3) STATISTICAL INFERENCES. 2 Hypothes is testing Hypothesis testing for population mean One population mean Two population mean Hypothesis

State the null hypothesis and alternative hypothesis

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Page 17: 1 CHAPTER 4 (PART 3) STATISTICAL INFERENCES. 2 Hypothes is testing Hypothesis testing for population mean One population mean Two population mean Hypothesis

State the null hypothesis and alternative hypothesis

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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.

Page 18: 1 CHAPTER 4 (PART 3) STATISTICAL INFERENCES. 2 Hypothes is testing Hypothesis testing for population mean One population mean Two population mean Hypothesis

State the null hypothesis and alternative hypothesis

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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?

Page 19: 1 CHAPTER 4 (PART 3) STATISTICAL INFERENCES. 2 Hypothes is testing Hypothesis testing for population mean One population mean Two population mean Hypothesis

Rejection region : reject H0

Level of significance,

Tails of a Test

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Two-Tailed Test

Left-Tailed Test

Right-Tailed Test

Sign in =

Sign in < >

Rejection Region In both tail In the left tail In the right tail

0H

1H

2. Determine the rejection and non-rejection regions

Page 20: 1 CHAPTER 4 (PART 3) STATISTICAL INFERENCES. 2 Hypothes is testing Hypothesis testing for population mean One population mean Two population mean Hypothesis

1. If the alternative hypothesis, H1 contains the less- than inequality symbol (<), the hypothesis is a left-tailed test.

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Reject H0

Accept H0

z

Reject H0 if Z < - Zα or p-value < α

Page 21: 1 CHAPTER 4 (PART 3) STATISTICAL INFERENCES. 2 Hypothes is testing Hypothesis testing for population mean One population mean Two population mean Hypothesis

2. If the alternative hypothesis, H1 contains the greater- than inequality symbol (>), the hypothesis is a right-tailed test.

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Reject H0

Accept H0

Reject H0 if Z > Zα or p-value < α

z

Page 22: 1 CHAPTER 4 (PART 3) STATISTICAL INFERENCES. 2 Hypothes is testing Hypothesis testing for population mean One population mean Two population mean Hypothesis

3. If the alternative hypothesis, H1 contains the not-equal-to symbol (≠), the hypothesis is a two-tailed test.

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Reject H0

Reject H0

Accept H0

Reject H0 if Z < Zα/2 or Z > Zα/2

2/z 2/z

Reject H0 if p-value < α/2

Page 23: 1 CHAPTER 4 (PART 3) STATISTICAL INFERENCES. 2 Hypothes is testing Hypothesis testing for population mean One population mean Two population mean Hypothesis

a) Population Mean, µ

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• is known

• is unknown and

• is unknown and

xZ

n

xZ sn

1

xt sn

v n

3. Calculate the value of test statistic

30n

30n

Page 24: 1 CHAPTER 4 (PART 3) STATISTICAL INFERENCES. 2 Hypothes is testing Hypothesis testing for population mean One population mean Two population mean Hypothesis

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1 2

For two large and independent samples

and and   are known.

1 2 1 2

2 21 2

1 2

x xZ

n n

1 2

1 2

For two large and independent samples

and and   are unknown.

(Assume )

1 2 1 2

2 21 2

1 2

x xZ

s sn n

1 2 1 2

1 2

2 21 1 2 2

1 2

1 1

1 1with

2

p

p

x xZ

Sn n

n s n sS

n n

1 2

1 2

For two large and independent samples

and and   are unknown.

(Assume )

b) Difference between two population means, 1 2

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1 2 1 2

2 21 2

1 2

22 21 2

1 22 22 2

1 2

1 1 2 2

1 11 1

x xt

s sn n

s sn n

vs s

n n n n

1 2 1 2

1 2

1 2

1 1

2

p

x xt

Sn n

v n n

1 2

1 2

For two small and independent samples

and and   are unknown.

(Assume )

1 2

1 2

• For two small and independent samples   

taken from two normally distributed  

populations. and   are unknown.

(Assume )

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c) Population proportion, p

d) Difference between two population proportions, 1 2p p

Page 27: 1 CHAPTER 4 (PART 3) STATISTICAL INFERENCES. 2 Hypothes is testing Hypothesis testing for population mean One population mean Two population mean Hypothesis

EXAMPLE 4.1

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SOLUTION

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1. H0 : µ ≤ 2400 (the claim is true) H1 : µ > 2400 (the claim is false)

2. We reject H0 if Z > 2.3263 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.

3263.2

xZ sn

97.18

40

40024003600

Page 29: 1 CHAPTER 4 (PART 3) STATISTICAL INFERENCES. 2 Hypothes is testing Hypothesis testing for population mean One population mean Two population mean Hypothesis

EXAMPLE 4.2

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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.

Page 30: 1 CHAPTER 4 (PART 3) STATISTICAL INFERENCES. 2 Hypothes is testing Hypothesis testing for population mean One population mean Two population mean Hypothesis

SOLUTION

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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 > 2.3263 or p-value < 0.01 (right tail test)

3. Test statistic, 1 2 1 2

2 21 2

1 2

x xZ

n n

6.1

3515

3010

0)495500(22

Page 31: 1 CHAPTER 4 (PART 3) STATISTICAL INFERENCES. 2 Hypothes is testing Hypothesis testing for population mean One population mean Two population mean Hypothesis

4. Since Z = 1.6 < 2.3263, we failed to reject H0 (accept H0)./ Since p-value= 0.0548 > 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.

33.26.101.0

Page 32: 1 CHAPTER 4 (PART 3) STATISTICAL INFERENCES. 2 Hypothes is testing Hypothesis testing for population mean One population mean Two population mean Hypothesis

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.

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Page 33: 1 CHAPTER 4 (PART 3) STATISTICAL INFERENCES. 2 Hypothes is testing Hypothesis testing for population mean One population mean Two population mean Hypothesis

SOLUTION

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1. H0 : p ≥ 0.95 (the claim is true/ conformed to specifications) H1 : p < 0.95 (the claim is false/ not conformed to specifications)

2. We reject H0 if Z < -2.33 or p-value < 0.01 (left tail test)

3. Test statistic,

4. Since Z = -55.8 < -2.33, we reject H0. We can conclude that the machine equipment supplied by the manufacturer is not conformed to specifications

8.55

200)05.0(95.0

95.009.0

33.2

Page 34: 1 CHAPTER 4 (PART 3) STATISTICAL INFERENCES. 2 Hypothes is testing Hypothesis testing for population mean One population mean Two population mean Hypothesis

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.

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Page 35: 1 CHAPTER 4 (PART 3) STATISTICAL INFERENCES. 2 Hypothes is testing Hypothesis testing for population mean One population mean Two population mean Hypothesis

SOLUTION

35

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 < -1.96 or Z > 1.96 (two tailed test)

095.0200

19ˆ

1 p

05.0100

2 p

Page 36: 1 CHAPTER 4 (PART 3) STATISTICAL INFERENCES. 2 Hypothes is testing Hypothesis testing for population mean One population mean Two population mean Hypothesis

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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.

35.1

)100

12001

)(92.0(08.0

0)05.0095.0(

08.0100200

519ˆ

p