MFGE 341Quality Science Statistics

What do we remember?

The Next Step

• Up to this point, we have only been able to estimate a population parameter.

• Now, we will start testing a previously made claim about a population parameter

Hypothesis Tests

• A process that uses sample statistics to test a claim about the value of a population parameter

• To test is a hypothesis is wrong, you first have to assume it is right.– It is innocent until proven guilty

Stating the Hypothesis

• The first step in testing a hypothesis is being able to state it correctly.– In addition, you need to state the complement to

the hypothesis• When one is false, the other is true

• The two statements are the null hypothesis and the alternative hypothesis

• ALWAYS ASSUME THE NULL HYPOTHESIS IS TRUE

Types of Error

• Type I error– The null hypothesis is rejected, but it is true

• Type II error– The null hypothesis is not rejected, but it is false

Level of Significance

• We need to determine our maximum allowable probability of making a type I error– We call this the level of significance & denote it α

• If α is small, then the probability of rejecting a true null hypothesis is also small

P-values

• A P-value in a hypothesis test is the probability of obtaining a sample statistic with a value as extreme or more extreme than the one determined from the sample data

• If the P-value is smaller than α then reject the null hypothesis

Finding P

• In order to find P, we first need to determine what type of test we are doing

• To determine the test type, look at the inequality in the alternative hypothesis

7 Steps

• State the hypotheses• Specify the level of significance• Determine the type of test• Calculate the test statistic• Find P• Decide whether to reject the null hypothesis• Interpret your answer

The claim is the null hypothesis

The claim is the alternate hypothesis

Reject the null hypothesis

There is enough evidence to reject the claim

There is enough evidence to support the claim

Accept the null hypothesis

There is not enough evidence to reject the claim

There is not enough evidence to support the claim

Test for the Mean

• Do we know the population standard deviation?– If the answer is yes, we can use a Z-test to

determine our P-value• Z=sample mean-population mean/

standard error

Requirements

• We need to approximate a normal distribution– Population needs to be normally distributed or

Sample Size needs to be at least 30– Sample needs to be random– Population standard deviation is known

Performing the Test

• We first make sure the requirements are satisfied• Next we perform the steps for hypothesis testing– When we need to find P, we use the Z value– Remember that P is twice the value that Z gives us

with a two tailed test

• If P is less than or equal to α we reject• If P is greater than α we fail to reject

Rejection Regions

• Another method of hypothesis testing is to determine the range of values for which the null hypothesis is rejected.– These are called rejection regions

• These regions have critical values which form a boundary between the rejection area and the non rejection area

Using Rejection Regions

• Go through the same process as before, except– Instead of finding the area that corresponds to a Z

value, we draw a graph and shade the rejection region

• If our Z value falls in the shaded area, we reject the null hypothesis

What is σ is Unknown

• If the population standard deviation is not known, we need to use a t-test instead of the z-test.

• Remember d.f.=n-1

Standard Deviation/Variance

• When testing measures of variance, we need to use the chi-square test with the following test statistic

• We need a random sample and normally distributed population

Proportions

• If we are testing a proportion (think binomial distribution) then we can use a Z-test with the following test statistic

• As long as np and nq are at least 5