Chapter 9 Hypothesis Testing

-- Hypothesis testing can be used to determine whethera statement about the value of a population parametershould or should not be rejected.

Null Hypothesis (H0 )

The null hypothesis, denoted by H0, is a tentativeassumption about a population parameter.

Alternative Hypothesis (Ha)

The alternative hypothesis, denoted by Ha, is theopposite of what is stated in the null hypothesis.
The alternative hypothesis is what the test isattempting to establish.

Hypothesis testing procedures uses data from a sample to test the null hypothesis and alternative hypothesis

Section 9.1 Developing Null and Alternative Hypothesis

Structure of Null and Alternative Hypothesis for 3 types of scenarios involving hypothesis testing

I] Testing Research Hypothesis

-- research hypothesis should be expressed as alternative hypothesis

-- The conclusion that the research hypothesis is truecomes from sample data that contradict the nullhypothesis.

Example: p 353 #2

H0:μ14

Ha:μ14

Conclusion is either H0is rejected or not rejected

II] Testing the Validity of a Claim

-- Manufacturers’ claims are usually given the benefitof the doubt and stated as the null hypothesis.

-- The conclusion that the claim is false comes fromsample data that contradict the null hypothesis.

Example: p 353 #1

H0: μ $600

Ha: μ > $600

Conclusion is either H0is rejected or not rejected

III] Testing in Decision-Making Situation

-- A decision maker might have to choose betweentwo courses of action, one associated with the null

hypothesis and another associated with thealternative hypothesis.

Example: p 353 #3

H0: μ = 32

Ha: μ= 32

Conclusion is either H0is rejected or not rejected

Summary of Forms for Null and Alternative Hypotheses about a Population Mean

-- The equality part of the hypotheses always appearsin the null hypothesis.

-- In general, a hypothesis test about the value of apopulation mean  must take one of the following

three forms (where 0 is the hypothesized value ofthe population mean).

One Tailed One Tailed Two-Tailed

(lower-tailed) (upper-tailed)

Example: Metro EMS

A major west coast city providesone of the most comprehensiveemergency medical services inthe world.

Operating in a multiplehospital system with approximately 20 mobile medicalunits, the service goal is to respond to medicalemergencies with a mean time of 12 minutes or less.

The director of medical serviceswants to formulate a hypothesistest that could use a sample ofemergency response times todetermine whether or not theservice goal of 12 minutes or lessis being achieved.

Null and Alternative Hypotheses

H0:  12 The emergency service is meeting the response goal; no

follow-up action is necessary.

Ha: > 12 The emergency service is notmeeting the response goal;appropriate

follow-up action isnecessary.

where:  = mean response time for the populationof medical emergency requests

Section 9.2 Type I and Type II Errors

-- When you reject or not reject the null hypothesis during the hypothesis testing procedures, the correct conclusions are not

always possible

-- Because hypothesis tests are based on sample info, we must allow for the possibility of errors

Type 1 Error

-- A Type I error is rejecting H0 when it is true.

-- The probability of making a Type I error when thenull hypothesis is true as an equality (i.e. μ = 12) is called thelevel of significance.

Level of Significance () probability of rejecting H0: μ 12 when the null hypothesis is true
In hypothesis testing, a person will specify the level of a type I error with common values  = .05 and  = .01
If the cost of a Type I error is high  small values of  are preferred. If the cost of making a type I error are low higher values of  are preferred

-- Applications of hypothesis testing that only controlthe Type I error are often called significance tests.

Type II Error

-- A Type II error is accepting H0 when it is false.

-- It is difficult to control for the probability of makinga Type II error.

-- Statisticians avoid the risk of making a Type IIerror by using the phrase “do not reject H0” and not the phrase “accept H0”.

Section 9.3 Population Mean: σ known

-- σ known involves having historical data or other info available that allows us to obtain a good estimate of the population standard deviation prior to sampling

-- using to perform hypothesis tests about population mean

We will be using the characteristics of the normal distribution to perform hypothesis testing of the population mean. With this assumption, keep in mind the following for the sampling distribution of and assumptions for a normal distribution: If population is:

1)normally distributed  any sample size is ok (Central Limit Theorem)

2)symmetric  sample size as small as 15 is ok

3)believed to be approximately normal  sample size of less than 15 is ok

4) not known or not normal sample size of at least 30 unless highly skewed then 50

I] One Tailed Tests

Lower-Tailed TestUpper Tailed Test

Test Statistic (Use of Standard Normal Random Variable)

-- use of the standard normal random variable z to determine whether deviates from the hypothesized value enough to justify rejecting the null hypothesis

A) p-Value Approach to One-Tailed Hypothesis Testing

-- The p-value is the probability, computed using the test statistic, that measures the support (or lack ofsupport) provided by the sample for the nullhypothesis. Also called the observed level of significance.

Smaller p-values are more evidence against H0

-- If the p-value is less than or equal to the level ofsignificance (), the value of the test statistic is in therejection region.

-- Reject H0 if the p-value  .

Lower-Tailed Test About a Population Mean: σ known

Upper-Tailed Test About a Population Mean: σ known

B) Critical Value Approach to One-Tailed Hypothesis Testing

--The test statistic z has a standard normal probabilitydistribution.

-- We can use the standard normal probabilitydistribution table to find the z-value with an areaof  in the lower (or upper) tail of the distribution.

-- The value of the test statistic that established theboundary of the rejection region is called thecritical value for the test.

-- The rejection rule is:

• Lower tail: Reject H0 if z -z

• Upper tail: Reject H0 if zz

Lower-Tailed Test About a Population Mean: σ known

Upper-Tailed Test About a Population Mean: σ known

Steps of Hypothesis Testing:

Step 1. Develop the null and alternative hypotheses.

Step 2. Specify the level of significance .

Step 3. Collect the sample data and compute the test statistic.

With p-Value Approach…

Step 4. Use the value of the test statistic to compute thep-value.

Step 5. Reject H0if p-value a.

With Critical Value Approach…

Step 4. Use the level of significanceto determine the critical value and the rejection rule.

Step 5. Use the value of the test statistic and the rejectionrule to determine whether to reject H0.

Example of One Tailed Test: Metro EMS

The response times for a randomsample of 40 medical emergencieswere tabulated. The sample meanis 13.25 minutes. The populationstandard deviation is believed tobe 3.2 minutes.

The EMS director wants toperform a hypothesis test, with a.05 level of significance, to determinewhether the service goal of 12 minutes or less is beingachieved.

II] Two Tailed Tests

General Form

-- used in situations when you are interested in whether the population mean deviates in either direction of a certain value

A) p-Value Approach toTwo-Tailed Hypothesis Testing

Compute the p-value using the following three steps:

1. Compute the value of the test statistic z.

2. If z is in the upper tail (z > 0), find the area underthe standard normal curve to the right of z.

If z is in the lower tail (z < 0), find the area underthe standard normal curve to the left of z.

3. Double the tail area obtained in step 2 to obtainthe p –value.

The rejection rule: Reject H0 if the p-value  .

B) Critical Value Approach to Two-Tailed Hypothesis Testing

-- The critical values will occur in both the lower andupper tails of the standard normal curve.

-- Use the standard normal probability distributiontable to find z/2 (the z-value with an area of a/2 in

the upper tail of the distribution).

The rejection rule is:Reject H0 if z -z/2or zz/2.

Example of a Two-Tailed Test: Glow Toothpaste

The production line for Glow toothpasteis designed to fill tubes with a mean weightof 6 oz. Periodically, a sample of 30 tubeswill be selected in order to check thefilling process.

Quality assurance procedures call forthe continuation of the filling process if thesample results are consistent with the assumption thatthe mean filling weight for the population of toothpastetubes is 6 oz.; otherwise the process will be adjusted.

Assume that a sample of 30 toothpastetubes provides a sample mean of 6.1 oz. The population standard deviation is

believed to be 0.2 oz.

Perform a hypothesis test, at the .03level of significance, to help determinewhether the filling process should continue

operating or be stopped and corrected.

Section 9.4 Population Mean: σ unknown

--is used as an estimate of μ and the sample standard deviation (s) is used as an estimate of σ

-- use the t distribution

Test Statistic

with n -1 degrees of freedom

Rejection Rule: p-value approach

Reject H0 if p-value 

Rejection Rule: critical value approach

H0: Reject H0 if t -t

Ha: 

H0: Reject H0 if t t

Ha: 

H0:  Reject H0 if t t/2 or H0 if t -t/2

Ha:  

P-Value and the t distribution

-- The format of the t distribution table provided in moststatistics textbooks does not have sufficient detailto determine the exact p-value for a hypothesis test.

-- However, we can still use the t distribution table to identify a range for the p-value.

-- An advantage of computer software packages is thatthe computer output will provide the p-value for the t distribution.

Example: One Tail Test σ unknown

A State Highway Patrol periodically samplesvehicle speeds at various locationson a particular roadway. The sample of vehicle speedsis used to test the hypothesis: H0: μ 65

The locations where H0 is rejected are deemedthe best locations for radar traps. At Location F, a sample of 64 vehicles shows a

mean speed of 66.2 mph with astandard deviation of4.2 mph. Use a = .05 totest the hypothesis.

Example 2 Tail Test σ unknown

A sample of 48 provided a sample mean of = 17 and s = 4.5. Use the following hypothesis and  = .05 to perform a 2 tail hypothesis test.

H0:  18

Ha:  18