1

INTRODUCTION TO HYPOTHESIS

TESTING

2

PURPOSE

A hypothesis test allows us to draw conclusions or make decisions regarding population data from sample data.

3

The Logic of Hypothesis Tests Assume a population distribution with a specified

population mean. State the hypothesized population mean (this

statement is referred to as the null hypothesis). Draw a random sample from the population and

calculate the sample mean. Determine the “relative position” on the calculated

mean on the distribution of sample means. If the sample mean is “close” to the specified population mean, we do not have evidence to reject the hypothesized population mean.

If the calculated sample mean is “not close” to the specified population mean, we conclude that our sample could not have been drawn from the hypothesized distribution, and thus, wereject the null hypothesis.

4

Example The president of City Real Estate claims that

the mean selling time of a home is 40 days after it is listed. A sample of 50 recently sold homes shows a sample mean of 45 days with a standard deviation of 20 days. Is the president correct?

5

ONE SAMPLE HYPOTHESIS TESTS

Applied to determine if the population mean is consistent with a specified value or standard

Two tests the z- test the t-test

6

ONE SAMPLE HYPOTHESIS TEST

Large Sample

Sample size: n>25

Null and Alternative Hypothesis = #Ha: =/ # or > # or < #

7

ASSUMPTIONS: z-TEST

the underlying distribution is normal or the Central Limit Theorem can be assumed to hold

the sample has been randomly selected

the population standard deviation is known or the sample size is at least 25.

clt 8

ExampleA manufacturer of electric ovens purchases

components with a specified heat resistance of 8000. A sample of 36 components selected from a large shipment shows an average heat resistance of 7900 and a standard deviation of 200. Can the manufacturer conclude that the heat resistance of the glass components is less than 8000?

9

ONE SAMPLE HYPOTHESIS TESTSmall Samples

Null and Alternative Hypothesis

= #Ha: =/ # or > # or < #

10

ASSUMPTIONS: t-TEST The underlying distribution is normal or the

CLT can be assumed to hold The samples have been randomly and

independently selected from two populations The variability of the measurements in the two

populations is the same and can be measured by a common variance. (There is a t-test that does not make this assumption; it is available when using Minitab.)

11

EXAMPLEA manufacturer uses a bottling process and

will lose money if the bottles do not contain the labeled amount. Suppose a cola company labels the bottles as 20 oz. A sample of 16 bottles results in 19.6 oz and a standard deviation of 0.3 oz. Does the process need an adjustment?

12

Paired Samples Test

Find the difference in the paired values Treat the difference scores as one sample. Apply a one sample test.

13

EXAMPLE

14

TWO-SAMPLES HYPOTHESIS TESTS

Applied to compare the values of two population means.

clt 15

The Distribution of the Difference Between Two Independent Samples

.SE as

estimated becan SE and normal,ely approximat is of

ondistributi sampling thelarge, are sizes sample theIf .3

.SE is ofdeviation standard The 2.

means. population the

in difference the, is ofmean The 1.

2

22

1

21

21

2

22

1

21

21

2121

n

s

n

s

xx

nnxx

xx

16

HYPOTHESIS TEST: TWO INDEPENDENT SAMPLESLarge Samples

Sample Size: n < 25

Null and Alternative Hypothesis

= Ha: =/ or > or <

17

HYPOTHESIS TEST: TWO INDEPENDENT SAMPLESSmall Samples

Null and Alternative Hypothesis

= Ha: =/ or > or <

clt 18

ExampleTwo machines are used in the manufacturer of steel

rings. The quality control director wishes to know if she should conclude machine A is producing rings with a different inside diameter than those produced by machine B.

Type A Type B

N 40 40

Mean 2” 1.5”

Variance 0.001” 0.002”

clt 19

Example/ProportionSports car owners complain that their cars

are judged differently from sedans at the vehicle inspection station. Previous records indicate that 30% of all cars fail inspection on the first time. A random sample of 150 sports cars produced 60 that failed. Is there a different standard?

clt 20

Estimating the Difference in Population MeansFor large samples, point estimates and their margin of error as well as confidence intervals are based on the standard normal (z) distribution.

2

22

1

21

2/21

21

)(

:-for interval Confidence

n

s

n

szxx

clt 21

Example/ProportionsIn producing a particular component, the

Shelby Co. has a defective rate of 2%. In a sample of 500, a contractor found a rate of 1%. Has the quality improved?