SPECIFIC OBJECTIVES:At the end of the unit, the students are expected to;1.Define hypothesis and hypothesis testing;2.Explain the type I (alpha) and type II (beta) errors in

rejecting or accepting the null hypothesis;3.Distinguish directional from non-directional

hypothesis;4.Determine the level of significance in a one-tailed or

two-tailed test;5.Apply an appropriate statistical test for the

hypothesis; and6.Summarize the procedure for hypothesis testing.

Definitions:• HYPOTHESIS is simply a statement that

something is true. It is a tentative explanation, a claim or assertion about people, objects or events.

• HYPOTHESIS is a form of statement and the truth/validity or certainty of any statement is questionable. It is imperative that such a statement must be tested significantly in order to ascertain its truth/validity.

Examplee.g. 1. “There is no significant relationship

between the mathematics attitude and competency levels of 2nd year accountancy students of the University of the East.”

e.g. 2. “The proportion of consumers who purchased Ariel powder soap before advertising campaign in the television and the proportion who purchased it after the advertising campaign is not equal.”

Some types of questions that are commonly asked:Is there significant difference between the

performance of UE graduates in the Oct CPA Board Exam and May CPA Board Exam?

Is there a significant difference in the proportion of consumers who purchased Ariel powder soap before advertising campaign in television and the proportion who purchased it after the advertising campaign?

Is there a significant difference in the mean life span between the Eveready and National batteries?

HYPOTHESIS TESTING: (Def)It is a procedure in making decisions based

on a sample evidence or probability theory used to determine whether the hypothesis is accepted or rejected. If the statement is found reasonable, then the hypothesis is accepted otherwise rejected.

Two (2) types of Hypothesis:1. NULL HYPOTHESIS (Ho)

Ho = Ha no difference/no relationship

“There is no significant relationship between the mathematics attitude and competency levels of 2nd yr accountancy students of the Univ. of the East.”

2. ALTERNATIV E HYPOTHESIS(Ha)Ho > HaHo < HaHo ≠ Ha

there is difference/relationship

E.g.“The proportion of consumers who purchased Ariel powder soap before advertising campaign in the television and the proportion who purchased it after the advertising campaign is not equal.”

The 4 Possible Outcomes for a Hypothesis Test

Decision/Fact

Ho is true Ho is false

Do not reject Ho

Reject Ho

Correct Decision

Type 1 error

Type 2 error

Correct decision

Def of Type 1 and Type 2 errors:I. Type 1 error: Rejecting the null hypothesis

when in fact the null hypothesis is true.

I. Type 2 error: Not rejecting the null hypothesis when in fact the null hypothesis is false.

Definition of Level of Significance:The significance level (α) of a hypothesis test

is defined to be the probability of committing a type 1 error. This is the probability of rejecting the null hypothesis.

The significance level (β) of a hypothesis test is defined to be the probability of committing a type 2 error. This is the probability of not rejecting a false or null hypothesis.

One-tailed and two-tailed test:• A one-tailed test is a hypothesis test for which

the rejection region lies at only one tail of the distribution.left-tailed test is when the population mean (μ) is less that the specified value μo. right-tailed test is when the population mean (μ) is greater than the specified value of μo.

• A two-tailed test is used when the alternative hypothesis is non-directional which means that the values of two measure of the same kind are not equal.

Rejection regions of one-tailed & two-tailed tests:

Two-tailed test

One-tailed test (Right)

One-tailed test (left)

Sign in Ha

Rejection region

≠

Both sides

>

Right side

<

Left side

Critical Region in Testing HypothesisLevel of Significance

Type of test

One-tailed test Two-tailed test

α = 0.05

α = 0.01

α = 0.10

Left-tailedZ < -1.645

Z < -2.33

Z < -1.28

Right – tailedZ > 1.645

Z > 2.33

Z > 1.28

Z > 1.96 or Z < -1.96Z > 2.575 or Z < -2.575Z > 1.645 or Z < -1.645

Note:Reject the null hypothesis when the

computed value of (z) lies within the area of rejection

Some Terminologies To Remember:• Test Statistic: The statistic used as a basis for

deciding whether the null hypothesis should be rejected.

• Rejection Region: The set of values of the test statistic that leads to rejection of the null hypothesis.

• Non-rejection Region: The set of values of the test statistic that leads to non-rejection of the null hypothesis.

• Critical value: The values of the test statistic that separate the rejection and non-rejection regions.

A Hypothesis Testing Procedure:1. Formulate the null and alternative

hypothesis.2. Decide the level of significance, α.3. Choose the appropraite test statistic.4. Establish the critical region.5. Compute the value of the statistical test.6. Decide whether to accept or reject the null

hypothesis.7. Draw a conclusion.

2 Categories involved in testing hypothesis between means:

• n ≥ 30; large sample• N < 30; small sample

USES OF z – Test and t – test:The z-test is used in comparing two means if the

population standard deviation (δ) is known. We should give emphasis in the discussion that if the population is normally distributed, z-test can be used for any sample size (n). However, in many practical cases, the population standard deviation is unknown but the sample is sufficiently large, that is n ≥ 30. The sample standard deviation (s) is used as an estimator of the population standard deviation.

SAMPLE PROBLEMS:1. The mean weight of the baggage

carried into an airplane by individual passengers at Tuguegarao City airport is 19.8 kgs. A statistician takes a random sample of 110 passengers and obtains a sample mean weight of 18.5 kgs with standard deviation of 8.5 kgs. Test the claim at α = 0.01 level of significance.

Procedure:1. Formulate the null and alternative

hypotheses:Ho: μ = 19.8 kgHa: μ = 19.8 kg

2. α = 0.01

3. The alternative hypothesis is expressed in a directional statement, therefore use one-tailed test.

Continuation: procedure4. The tabular or critical value of z = -2.33.

5. Compute the z= value.Given: mean = 18.5 kg

μ = 19.8 kg s = 8.5 kg n = 110

HYPOTHESIS ABOUT MEANS: Z = ( x – μ)/s/√n

= (18.5 – 19.8)/8.5/ √110= -1.60

Cont.6. The computed value of z = -1.60 lies under

the non-rejection area, therefore accept the null hypothesis (Ho).

7. Conclusion:There is no significant difference

between the weight of baggage carried by individual passengers.

Other Test Statistic:ii. Difference between means:

z = (mean₁ - mean₂)/ √ s₁/n ₁ + s₂/n₂

iii. Hypothesis testing about a single proportion:z = (p` - p)/√p(1 – p)/n p` ; sample proportionp ; population proportionn ; number of cases

iv. Hypothesis Testing About Two Proportions

Test Concerning Means:• Example:An agronomist randomly selected 20 matured

calamansi trees of one variety and have a mean height of 10.8 ft with standard deviation of 1.25 ft, while 12 randomly selected calamansi trees of another variety have a mean height of 9.6 ft with standard deviation of 1.45 ft. Test whether the difference between the two sample means is significant. Use α = 0.05.

PROCEDURE:1. Ho: mean₁ = mean₂

Ho: mean₁ ≠ mean₂2.

3. The alternative hypothesis is non-directional, thus, the two-tailed test is used.

4. Since there are two samples used.df = n₁ + n₂ - 2 = 30

α = 0.05

Cont.5. Compute the t – value:

Given: x₁ = 10.8 ft s₂ = 1.45 ft x₂ = 9.6 ft n₁ = 20 s₁ = 1.25 ft n₂ = 12

6. Formula:t = (x₁ - x₂)/ √s₁

2/n ₁ + s₂2/n₂

= (10.8 – 9.6)/ √(1.25)2/20 + (1.45) 2/12t = 2.38

Cont.7. The computed value of t = 2.38 is greater

than the tabular value of t = 2.042, thus, reject the null hypothesis (Ho) and accept the alternative hypothesis (Ha).

8. Conclusion:There is a significant difference between the two samples.

Assignment:1. The hospital record shows that the

mean weight of newly born baby is 7 lbs, with the standard deviation of 0.75 lbs. A researcher takes a sample of 55 newly born babies and found to have a mean weight of 6.73 lbs. Test the claim at 0.05 level of significance.