10.2 Hypothesis Testing
HYPOTHESIS is a statement about a population parameter subject to verification.
Data are then used to check the reasonableness of the statement. To begin, we need to define the word hypothesis.
Overview:
Laws: In the legal system, a person is innocent until proven guilty. A jury hypothesizes that a person charged with a crime is innocent and subjects this hypothesis to verification by reviewing the evidence and hearing testimony before reaching a verdict.
Medical field: In a similar sense, a patient goes to a physician and reports various symptoms. On the basis of the symptoms, the physician will order certain diagnostic tests, then, according to the symptoms and the test results, determine the treatment to be followed.
In statistical analysis, we make a claim—that is, state a hypothesis—collect data, and then use the data to test the assertion. We define a statistical hypothesis as follows.
HYPOTHESIS is a statement about a population parameter subject to verification.
In most cases, the population is so large that it is not feasible to study all the items, objects, or persons in the population. For example, it would not be possible to contact every teacher in the Cambodia to find his or her monthly income.
So, What is Hypothesis Testing?
The terms hypothesis testing and testing a hypothesis are used interchangeably. Hypothesis testing starts with a statement, or assumption, about a population parameter—such as the population mean.
Example:
This statement is referred to as a hypothesis. A hypothesis might be that the mean monthly commission of sales associates in retail electronics stores, such as Panasonics, is $2,000.
You want to test whether there is a relationship between gender and height.
You want to know, there is a relationship between mask sold and covid cases updates
HYPOTHESIS TESTING A procedure based on sample evidence and probability theory to determine whether the hypothesis is a reasonable statement.
Five-Step Procedure for Testing a Hypothesis
Step 1: State null and alternate hypothesis
Step 2: Select a level of significance
Step 3: Identify the test statistic
Step 4: Formulate a decision rule
Step 5: Take a sample, arrive at decision
Term:
Test Statistic is a value, determined from sample information, used to determine whether to reject or accept the null hypothesis.
CRITICAL VALUE: The dividing point between the region where the null hypothesis is rejected and the region where it is not rejected.
Step 1: State your null ( H0 ) and alternate hypothesis ( H1 )
The first step is to state the hypothesis being tested. It is called the null hypothesis, designated H0, and read “H sub zero.” The capital letter H stands for hypothesis, and the subscript zero implies “no difference.” There is usually a “not” or a “no” term in the null hypothesis, meaning that there is “no change.”
For example, the null hypothesis is that the mean number of miles driven on the steel-belted tire is not different from 60,000. The null hypothesis would be written H0: 60,000. Generally speaking, the null hypothesis is developed for the purpose of testing. We either reject or fail to reject the null hypothesis. The null hypothesis is a statement that is not rejected unless our sample data provide convincing evidence that it is false.
We should emphasize that, if the null hypothesis is not rejected on the basis of the sample data, we cannot say that the null hypothesis is true. To put it another way, failing to reject the null hypothesis does not prove that H0 is true, it means we have failed to disprove H0.
⚠️ To prove without any doubt the null hypothesis is true, the population parameter would have to be known. To actually determine it, we would have to test, survey, or count every item in the population. This is usually not feasible. The alternative is to take a sample from the population.
NULL HYPOTHESIS A statement about the value of a population parameter developed for the purpose of testing numerical evidence.
ALTERNATE HYPOTHESIS A statement that is accepted if the sample data provide sufficient evidence that the null hypothesis is false.
💡 Thing To Remember
Equality is always part of H0 (e.g. “=” , “≥” , “≤”), but “≠” “<” and “>” always part of H1
In problem solving, look for key words and convert them into symbols. Some key words include: “improved, better than, as effective as, different from, has changed, etc.”
Step 2: Select a Level of Significance
After setting up the null hypothesis and alternate hypothesis, the next step is to state the level of significance.
LEVEL OF SIGNIFICANCE: The probability of rejecting the null hypothesis when it is true
The level of significance is designated ⍺, the Greek letter alpha. It is also sometimes called the level of risk. This may be a more appropriate term because it is the risk you take of rejecting the null hypothesis when it is really true.
There is no one level of significance that is applied to all tests.
A decision is made to use the .05 level (often stated as the 5 percent level), the .01 level, the .10 level, or any other level between 0 and 1.
Traditionally, the .05 level is selected for consumer research projects, .01 for quality assurance, and .10 for political polling.
You, the researcher, must decide on the level of significance before formulating a decision rule and collecting sample data.
💡 Therefore, If you have a large sample size and low variance, you can use a lower significance level, such as 0.01 to increase the confidence in your results. If you have a small size and high variance, you may want to use higher significance level, such as 0.1, to increase the sensitivity of your test.
Step 3: Select the Test Statistic
There are many test statistics. In this chapter, we use both z and t as the test statistic because it is one sample tests of Hypothesis. In later chapters, we will use such test statistics as F and X^2, called chi-square.
TEST STATISTIC A value, determined from sample information, used to determine whether to reject the null hypothesis.
In hypothesis testing for the mean (μ) when 𝞂 is known or sample size is large ( n ≥ 30 ), the test statistic z is computed by:
Step 4: Formulate the Decision Rule
A decision rule is a statement of the specific conditions under which the null hypothesis is rejected and the conditions under which it is not rejected.
The region or area of rejection defines the location of all those values that are so large or so small that the probability of their occurrence under a true null hypothesis is rather remote.
Chart portrays the rejection region for a test of significance that will be conducted later in the chapter.
Step 5: Make a Decision
The fifth and final step in hypothesis testing is computing the test statistic, comparing it to the critical value, and making a decision to reject or not to reject the null hypothesis. Referring to Chart above, if, based on sample information, z is computed
Exercise:
Case: Testing for a Population Mean with a Known Population Standard Deviation
Eg.1: Jamestown Steel Company manufactures and assembles desks and other office equipment . The weekly production of the Model A325 desk at the Fredonia Plant follows the normal probability distribution with a mean of 200 and a standard deviation of 16. Recently, new production methods have been introduced and new employees hired. The VP of manufacturing would like to investigate whether there has been a change in the weekly production of the Model A325 desk. We chose 50 week as sample and sample mean is 203.5
Solution to e.g. 1:
Step 1: State the null hypothesis and alternative hypothesis
H0 : μ = 200
H1 : μ ≠ 200
💡 note: keyword in the problem “has changed”
Step 2: Select the level of significance
⍺ = 0.01 as stated in the problem
Step 3: Select the test statistic
Use Z-distribution since 𝞂 ( standard deviation of population ) is known and only one sample.
Step 4: Formulate the decision rule
This example is two tailed ( H1 has sign ≠ ) test and ⍺ = 0.01 ( Confidence Level: 99% ), so Z = 2.85 => Find the Z score in Z table.
Step 5: Make a decision and interpret the result
Because computed value of Z = 1.55 not > Z value in Table, so it does not fall in the rejection region, H0 is not rejected. We conclude that the population mean is not different from 200.
So we would report to the vice president of manufacturing that the sample evidence does not show that the production rate at the plant has changed from 200 per week.
Case: Testing for a Population Mean with a Known Population Standard Deviation
Eg.2: CSTAD Company manufactures and assembles desks and other office equipment . The weekly production of the Model C369 desk follows the normal distribution with a mean of 200 and a standard deviation of 16. Recently, new production methods have been introduced and new employees hired. The VP of manufacturing would like to investigate whether there are any modified in the weekly production of the Model C369 desk. Is the mean number of desks produced at CSTAD has increased from 200 at the 0.05 significance level? Please perform hypothesis testing with sample of 50 weeks with sample mean is 205.
Solution to e.g. 2
Step 1: State the null hypothesis and alternative hypothesis
H0 : μ ≤ 200
H1 : μ > 200
💡 note: keyword in the problem “has increased”
Step 2: Select the level of significance
⍺ = 0.05 as stated in the problem
Step 3: Select the test statistic
Use Z-distribution since 𝞂 ( standard deviation of population ) is known and only one sample.
Step 4: Formulate the decision rule
x = 205, μ = 200, 𝞂 = 16, n = 50
So we computed Z = 2.21
What about Z value of Z table, >> One Tailed Test ( sign > of H1) and Risk: 0.05, it mean Confidence Level: 95%, we can assume that Z score = 1.65
Step 5: Make a decision and interpret the result
Due to Z computed > Z table, so we don't have enough evidence to accept H0, It meant that there are increase from 200 if CSTAD use new methods and hired new employee.
Last updated
