Hypothesis Testing in Statistics and Probability, Lecture notes of Mathematics

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STATISTICS

PROBABILITY

WHERE AM I NOW?

Identify the region where each of the given values falls.

1. t = 1.
2. t = 0.
3. t = - 1. Region C Region B Region B 4. t = - 2. 5. t = 2. Region A Region C

HYPOTHESIS

❖ A hypothesis is a proposed explanation, assertion, or

assumption about a population parameter or about the

distribution of a random variable.

Examples

❖ Does the mean height of Grade 12 students differ from 66 inches? ❖ Is the proportion of senior male students’ height significantly higher than that of senior female students?

NULL HYPOTHESIS ❖ The null hypothesis is an initial claim based on previous analysis that the researcher tries to disprove, reject or nullify. ❖ It shows no significant difference, no changes, nothing happened, or no relationship between the two parameters. ❖ The independent variable has no effect on the dependent variable. ❖ It is denoted by 𝐻 𝑜 ❖ It can be written in symbol, 𝐻𝑜: 𝜇 1 = 𝜇 2

FORMULATING HYPOTHESIS

Null Hypothesis

Alternative Hypothesis

equal to , the same as , not changed from , is = less than , lower than , below , smaller than , at most < > greater than , higher than , above , bigger than , at least ≠ not equal , different from , changed from , not the same as

Example 1

State the null and alternative hypotheses.

The school record claims that the mean score in Math of

the incoming Grade 11 students is 81.

Null hypothesis: Alternative hypothesis: The mean score in Math of the incoming Grade 11 students is 81. The mean score in Math of the incoming Grade 11 students is not 81. 𝐻𝑜: 𝜇 = 81 𝐻𝑎: 𝜇 ≠ 81

Example 3

State the null and alternative hypotheses.

A nutritionist wants to determine whether the average

number of calories of a low-calorie meal is more than 300.

Null hypothesis: Alternative hypothesis: The average number of calories of a low-calorie meal is 300. The average number of calories of a low-calorie meal is more than 300. 𝐻𝑜: 𝜇 = 300 𝐻𝑎: 𝜇 > 300

State the null and the alternative hypotheses of the following statements. QUIZ #

LEVEL OF SIGNIFICANCE

The most common levels of significance used are

1 %, 5 %, and 10 %.

In symbol, it is written as:

If alternative hypothesis used ≠, then alpha will be divided by 2. 𝛼 2 = 0. 005 , 𝛼 2 = 0. 025 , 𝛼 2 = 0. 05

Example 1

Determine the value of 𝛼 or 𝛼 2 based on the alternative hypothesis.

Maria uses a 5 % level of significance in proving that

there is no significant change in the average number of

enrollees in the 10 sections for the last two years.

Example 3

Determine the value of 𝛼 or 𝛼 2 based on the alternative hypothesis. The average number of years to finish basic education is 14. A sample of 30 senior high school students were asked and found out that the mean number of years to finish their basic education is 12 with a standard deviation of 2 years. Test the hypothesis at 99 % confidence interval that the average number of years to finish basic education is less than 14 years.

TWO-TAILED TEST VS ONE-TAILED TEST

❖When the alternative hypothesis is two-sided like

𝑎

1

2

, it is called a two-tailed test.

❖When the given statistics hypothesis assumes a less than or

greater than value, it is called one-tailed test.

less than , lower than , below , smaller than < > greater than , higher than , above , bigger than ≠ not equal , different from , changed from , not the same as

• QUIZ #

ILLUSTRATION OF THE REJECTION REGION

❖ The rejection region (critical region) is the set of all values of the test statistic that causes us to reject the null hypothesis. ❖ The non-rejection region (acceptance region) is the set of all values of the test statistic that causes us to fail to reject the null hypothesis. ❖ The critical value is a point (boundary) on the test distribution that is compared to the test statistic to determine if the null hypothesis would be rejected.