Introduction Hypothesis testing for one mean Hypothesis testing for one proportion Hypothesis testing for two mean (difference) Hypothesis testing for two proportion (difference)
Hypothesis and Test ProceduresA statistical test of hypothesis consist of :1.2. Calculate test statistic or using p-value3. Find Critical value4. Determine rejection region5. Make a conclusion*We test a certain given theory / belief about population parameter.We may want to find out, using some sample information, whether or not a given claim / statement about population is true.Develop Hypothesis Statement (H0 & H1)
Hypothesis testing : can be used to determine whether a statement about the value of a population parameter should or should not be rejected.Null hypothesis, H0 : A null hypothesis is a claim (or statement) about a population parameter that is assumed to be true. (the null hypothesis is either rejected or fails to be rejected.)Alternative hypothesis, H1 : An alternative hypothesis is a claim about a population parameter that will be true if the null hypothesis is false.Test Statistic : is a function of the sample data on which the decision is to be based.p-value: is the probability calculated using the test statistic. The smaller the p-value, the more contradictory is the data to H0.
*Critical point: It is the first (or boundary) value in the critical region.Rejection region: It is a set of values of the test statistics for which the null hypothesis will be rejected.
It is not always obvious how the null and alternative hypothesis should be formulated.
When formulating the null and alternative hypothesis, the nature or purpose of the test must also be taken into account. We will examine:The claim or assertion leading to the test.The null hypothesis to be evaluated.The alternative hypothesis.Whether the test will be two-tail or one-tail.A visual representation of the test itself.
In some cases it is easier to identify the alternative hypothesis first. In other cases the null is easier.
9.1.1 Alternative Hypothesis as a Research Hypothesis Many applications of hypothesis testing involve an attempt to gather evidence in support of a research hypothesis. In such cases, it is often best to begin with the alternative hypothesis and make it the conclusion that the researcher hopes to support. The conclusion that the research hypothesis is true is made if the sample data provide sufficient evidence to show that the null hypothesis can be rejected.
Example 9.1: A new drug is developed with the goalof lowering blood pressure more than the existing drug. Alternative Hypothesis H1 : The new drug lowers blood pressure more than the existing drug. Null Hypothesis H0 : The new drug does not lower blood pressure more than the existing drug.
9.1.2 Null Hypothesis as an Assumption to be Challenged We might begin with a belief or assumption that a statement about the value of a population parameter is true. We then using a hypothesis test to challenge the assumption and determine if there is statistical evidence to conclude that the assumption is incorrect. In these situations, it is helpful to develop the null hypothesis first.
Example 9.2 : The label on a soft drink bottle states that it contains at least 67.6 fluid ounces. Null Hypothesis H0 : The label is correct. > 67.6 ounces. Alternative Hypothesis H1 : The label is incorrect. < 67.6 ounces.
Example 9.3: Average tire life is 35000 miles. Null Hypothesis H0 : = 35000 miles Alternative Hypothesis H1: 35000 miles
Rule to develop H0 and H1:
Two-Tailed TestLeft-Tailed TestRight-Tailed TestH0== or = or H1
1. Coca Cola claim that on average, the cans contain less than 12 ounces of soda.
2. The mean family size in UK was 3.18 in 1998. Test whether the claim is true.
The mean starting salary of school teachers in Kangar is RM2800. Test whether the current mean starting salary of all teachers in Kangar is higher than RM2800.
The entire set of values that the test statistic may assume is divided into two regions. One set, consisting of values that support the and lead to reject , is called the rejection region. The other, consisting of values that support the is called the acceptance region. H0 always gets =.
Rule to Reject H0:
Tails of a Test
*9.1.3 How to decide whether to reject or accept ?
Two-Tailed TestLeft-Tailed TestRight-Tailed TestSign in == or = or Sign in Rejection RegionIn both tailIn the left tailIn the right tail
Because hypothesis tests are based on sample data, we must allow for the possibility of errors. A Type I error is rejecting H0 when it is true The probability of making a Type I error when the null hypothesis is true as an equality is called the level of significance (). Applications of hypothesis testing that only control the Type I error are often called significance tests.
A Type II error is accepting H0 when it is false. It is difficult to control for the probability of making a Type II error, . Statisticians avoid the risk of making a Type II error by using do not reject H0 and not accept H0.9.2.2 Type II Error
Type I and Type II ErrorsCorrectDecisionType II ErrorCorrectDecisionType I ErrorReject H0Do not reject H0
H0 FalseConclusionPopulation Condition
Null Hypothesis :
Test Statistic :
any population, is known and n is largeOr n is small any population, is unknown and n is large normal population, is unknown and n is small
Alternative hypothesisRejection Region ( Reject H0)Both tail: Right tail:Left tail:
Definition 9.2: p-value The p-value is the smallest significance level at which the null hypothesis is rejected.
Exercise:A teacher claims that the student in Class A put in more hours studying compared to other students. The mean numbers of hours spent studying per week is 25hours with a standard deviation of 3 hours per week. A sample of 27 Class A students was selected at random and the mean number of hours spent studying per week was found to be 26hours. Can the teachers claim be accepted at 5% significance level?
Alternative hypothesisRejection RegionBoth tail :Right tail:Left tail:
When working properly, a machine that is used to make chips for calculators does not produce more than 4% defective chips. Whenever the machine produces more than 4% defective chips it needs an adjustment. To check if the machine is working properly, the quality control department at the company often takes sample of chips and inspects them to determine if the chips are good or defective. One such random sample of 200 chips taken recently from the production line contained 14 defective chips. Test at the 5% significance level whether or not the machine needs an adjustment.
A manufacturer of a detergent claimed that his detergent is at least 95% effective is removing though stains. In a sample of 300 people who had used the detergent, n 279 people claimed that they were satisfied with the result. Determine whether the manufacturers claim is true at 1% significance level.
For two small and independent samples taken from two normally distributed populations.
Alternative hypothesisRejection Region
The mean lifetime of 30 batteries produced by company A is 50 hours and the mean lifetime of 35 bulbs produced by company B is 48 hours. If the standard deviation of all bulbs produced by company A is 3 hour and the standard deviation of all bulbs produced by company B is 3.5 hours.
Test at 1 % significance level that the mean lifetime of bulbs produced by Company A is better than that of company B (claim).
Alternative hypothesisRejection Region
A researcher wanted to estimate the difference between the percentages oftwo toothpaste users who will never switch to other toothpaste. In a sample of 500 users of toothpaste A taken by the researcher, 100 said that they will never switched to another toothpaste. In another sample of 400 users of toothpaste B taken by the same researcher, 68 said that they will never switched to other toothpaste.
At the significance level 1%, can we conclude that the proportion of users of toothpaste A who will never switch to other toothpaste is higher than the proportion of users of toothpaste B who will never switch to other toothpaste?
In a process to reduce the number of death due the dengue fever, two district, district A and district B each consists of 150 people who have developed symptoms of the fever were taken as samples. The people in district A is given a new medication in addition to the usual ones but the people in district B is given only the usual medication. It was found that, from district A and from district B, 120 and 90 people respectively recover from the fever.
Test the hypothesis that the new medication helps to cure the fever using a level of significance of 5% (claim).