Chapter 7 Statistics, Study notes of Mathematics

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CHAPTER 6

Statistical Hypothesis Testing

in Economics and Business

PhD. Vương Thị Thảo Bình

July 01, 2024

CASE STUDY

The Doll Computer Company makes its own computers and delivers them directly to customers who order them via the Internet. Doll competes primarily on speed of delivery. The computers are stored in the warehouses from which it generally takes 1 day to deliver a computer to the customer. This strategy requires high levels of inventory that add greatly to the cost. The operations manager to know whether the (population) mean of demand during lead time is different from 350 , which may be the point at which the current inventory policy needs to be changed. He notes that daily demand is random variable. He concludes that demand during lead time is normally distributed. He observes 25 lead time periods and records the demand during each period. These data are listed here. From long experience, the manager knows that the standard deviation is 75 computers. Is there enough evidence to conclude that the mean is not equal to 350?

Nonstatistical Hypothesis Testing (in a courtroom) In fact, the jury conducts a test of hypothesis. There are actually two hypotheses that are tested:

• The null hypothesis is H 0 : The defendant is innocent
• The alternative hypothesis is H 1 : The defendant is guilty

• The jury does not know which hypothesis is true. They must make a decision on the basis of evidence presented.
• The jury assume a man is innocent (H 0 is true) until proven guilty (until H 1 is shown based on evidence presented).

State the null hypothesis and alternative hypothesis Ex 1. A branch manager needs to check that the average queuing time for customers waiting to see a cashier is no more than 5 minutes. State the null and alternative hypotheses you will be using in this instance.

HW: State the null hypothesis and alternative hypothesis Ex 2. Test the hypothesis that at most 25 % of all members of a population conducted no marketing studies before opening their businesses. State the null and alternative hypotheses. Ex 3. Test the hypothesis that the population proportion is larger than one-half. Ex 4. Test the population variance for daily output does not exceed 500. Ex 5. Test that the population variance is at most 2 , 25.

HW: State the null hypothesis and alternative hypothesis Ex 6. A diet doctor claims that the average North American is more than 20 pounds overweight. To test his claim, State the null and alternative hypotheses. Ex 7. In some states, the law requires drivers to turn on their headlights when driving in the rain. A highway patrol officer believes that less than one- quarter of all drivers follow this rule. To test the claim, state the null and alternative hypotheses.

Hypothesis test decisions

Hypothesis test decision is a procedure to either

reject or accept (fail to reject) the null hypothesis

on the basis of sample evidence.

Ex

(4)

. Test the hypothesis that the population

mean is different from 5 minutes. Given that = 2

minutes; A sample of n=30, x =5.5.

Ex 1 ( 4 )

. Test the hypothesis that the population mean is different from 5 minutes. A random sample of 64 observations is obtained from a normally distributed population with mean  and = 2 minutes. Given that ; = 0. 05. H 0 :  =  0 = 5 H 1

0

• If H 0 is assumed to be TRUE is true then p(Z  W 

)=?  ZW

 or ZW x = 5. ( ) 5 0; X Z N n  − =  / 5 W ; 1. x Z Z z n       −  = (^)  =  =      5.5 5

• 2 W 2 64 ob z  − =  

Conclude that there is enough evidence

to support the alternative hypothesis.

Ex 1 ( 5 )

. Test the hypothesis that the population mean is different from 5 minutes. A random sample of 64 observations is obtained from a normally distributed population with mean  and = 2 minutes. Given that ; = 0. 05. H 0 :  =  0 = 5 H 1

0 x =5. / 5 W ; 1. x Z Z z n       −  = (^)  =  =      5.2 5

• 0.8 W 2 64 ob z  − =  ^ There is not enough evidence to support alternative hypothesis.

Conclusions of a Test of Hypothesis The goal of the process is to determine whether there is enough evidence to infer that the alternative hypothesis is true. If we reject the null hypothesis, we conclude that there is enough statistical evidence to infer that the alternative hypothesis is true. If we do not reject the null hypothesis, we conclude that there is not enough statistical evidence to infer that the alternative hypothesis is true.

Two possible errors can be made in any test

• The aim of statisticians is to minimise the probability of these errors.
• In our justice system, Type I errors are regarded as more serious.
• The significance level α is the probability of a Type I error.

As a consequence, the system is set up so that the probability of a Type I error is small.

REVIEW

• Library - Kahoot!

• is a rejection region of:

a) H

1

0

b) H

1

0

c) H

1

0 α α μ σ  (^) −  = (^)  =  −    0 X W Z ; Z z / n