hypothesis testing skills set

8/18/2019 Hypothesis Testing Skills Set

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Skills Set

Chapter 7: Hypothesis Testing

No. Skills Examples of questions involving the skills

1 To identify

the null and

alternativehypothesis

(If you

determine it’s

a one tail test:

look at sample

mean)

Lecture example 1 In the following situations, state suitable null and alternative hypotheses.

(a) The mean factory assembly time for a particular electronic component is

84 seconds. The factory manager wants to test whether the introduction

of a new assembly process results in a different assembly time.

(b) In a report, it was stated that the average age of all hospital patients was

53 years. The hospital director wants to test whether this figure is an

underestimate.

Solution

(a) Let be the mean factory assembly time (in seconds) after the

introduction of the new assembly process.

H0: 84 against H1: 84

(b) Let be the average age (in years) of the hospital patients.

H0 : 53 against H1 : 53 Tutorial Q4

A bank has branches in two cities A and B in the same country. The manager ofthe bank claims that the waiting time for a customer to open a new account

with the bank is not more than 5 minutes.

0H : 5 against 1H : 5

8235 486666667

150

ww

n

.

Note: 5w corresponds with 1H : 5

2 Using G.C (p-value) to

conduct

hypothesis

tests when no

unknowns are

present

Lecture example 2

A machine packs sand into bags. The mass of a filled bag follows anormal distribution with mean 1500 g and variance 0.16 g2. A sample of

10 filled bags is randomly chosen and found to have a mean mass of

1499.8 g.

(a) Test whether the sample provides significant evidence, at the 1%level of significance, that the machine packs bags that are

underweight.

Solution


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(a) Let X be the mass (in grams) of a filled bag.

Let be the mean mass (in grams) of the bags produced by the

machine.

0

1

H : 1500

H : 1500

n = 10, x = 1499.8, 0.16 0.4 .

Under H0,0.16

~ N 1500,10

X

.

Test statistic, Z 1500

~ N(0, 1)0.4

10

X under H0.

Using z -test, 1.58113883 z . p-value P( 1.58113883) Z = 0.0569.

Conclusion: Since p-value = 0.0569 > 0.01 , we do not reject H0 at

the 1% level of significance. There is insufficient evidence to saythat the machine packs bags that are underweight.

3 Using Test-Statistic and

Critical Region

when there are

unknowns in

the question.

Tutorial Q10b

A random sample of 50 packets of ‘Nilo Bar’ snacks is weighed and the mass,

x grams, is recorded. The results are summarised by

( x – 150) = –250, ( x – 150)2 = 3000.

The population mean mass is grams. A test is carried out at the 1%

significance level with the following hypotheses:

0 : H 0

:1 H 0

Given that 0 H is rejected in favour of H1, find the set of possible values of

0 .

Solution

x = 150

150 x

n

250 150 145

50

2 s =

2

2 1501150

1

x x

n n

2

25013000

49 50

35.714


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0Test statistic: ~ 0, 135.714

50

X Z N

Since0 H

is rejected in favour of H1,

Critical region: 2.576 or 2.576 z z

0

0

1452.576

35.714

50

147.18

0

0

1452.576

35.714

50

142.82

0 0 0: 142.82 or 147.18

4 Identify theneed to use

Central Limit

Theorem for

distribution of

sample mean.

(n large and

population not

known to be

normal)

Lecture example 5Clara, who has diabetes, has to monitor her blood glucose levels which vary

throughout a week. The results from a sample of 75 readings, x , taken at

random times over a week, are summarized by

511.5 x and 2 4027.89 x .(ii) State, giving a reason, whether we can conclude that the alternative

hypothesis0

is to be accepted if the blood glucose level cannot be

assumed to have a normal distribution.

Solution

(ii) Since the sample size, n, is sufficiently large, by the Central Limit

Theorem, the mean blood glucose level is approximately normally

distributed, so the alternative hypothesis0 can still be accepted.

6 conduct t-testsfor a

population

mean based on

a sample taken

from a normal

population with

unknown

variance and a

small sample

size.

Assumption

Lecture example 7An automatic coffee dispensing machine is set to dispense 440 cm

3 of coffee

into a mug. A random sample of 12 filled mugs was taken and the mugs were

found to contain the following amounts of coffee, in cm3.

420 446 372 430 436 441

433 422 445 422 438 442

Assuming no spillage and that the amount of coffee dispensed follows a normal

distribution, test at the 5% level of significance, whether the machine is

dispensing too little coffee.

Solution


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phrasing.

Using data lists

vs stats values.

Let be the mean amount of coffee in a mug (in cm3).

H0 : = 440

H1 : < 440

n = 12, x = 428.92, s = 20.07.

Under H0, test statistic, T = 440 X S

n

t (11).

Using t -test, t = –1.912418829.

p-value P( –1.912418829)T 0.0411.

Conclusion: Since p-value = 0.0411 < 0.05, we reject H0 at the 5% level of

significance. There is sufficient evidence to conclude that the machine is

dispensing too little coffee.

7 Comparing thedifferences

between 1 tail

and 2 tail test,

z test and t-

tests

Tutorial Question 7In a test at the 5% significant level it is found that there is significant evidence

that the population mean talk-time is less than 5 hours. Using only this

information, and giving a reason in each case, state whether each of the

following statements is

(i)

necessarily true, (ii) necessarily false,

(ii)

or (iii) neither necessarily true nor necessarily false.

(a) There is significant evidence at the 10% significance level that the

population mean talk-time is less than 5 hours.

(b) There is significant evidence at the 5% significance level that the

population mean talk-time is not 5 hours.

Solution

(a) Test at 10% level of significance: the p-value remains the same; hence if p-

value 0.05 , it is definitely 0.1 , and we reject 0H . This statement is

necessarily true.

(b) Test0H : 5 vs 1H : 5 at 5% level of significance: new p -value is

twice old p -value; new p-value may or may not be 0.05 . This statement is

neither necessarily true nor necessarily false.


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8 Comparing thedifferences

between z- test

and t-tests

Tutorial Question 8(b)After the implementation of another teaching pedagogy, the SAT score of 10

students are collected. To investigate the effectiveness of this pedagogy, a

hypothesis test is carried out at the %α level of significance.

(ii) If the null hypothesis is not rejected when a t -test is carried out, explain

whether it is necessarily true that the same conclusion is obtained when

a z-test is carried out at the same significance level.

Let andt z p p be the p-values obtained using a t -test and z-test

respectively. Note that for the same test statistic, >t z p p .

Since H0 is not rejected for a t -test, pt >100

. There are two possible

conclusions for a z-test.

OR

Z

N(0,1)

t( )

Test statistic

t p

z p

N(0,1)

t( )

z

Zt


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Reason:

For a fixed level of significance, , z

< t

(refer to graph above).

When H0 is not rejected for a t -test, there are two possible conclusions

for a z-test:

1) z

< test statistic < t

: H0 will be rejected if z-test is used.

2) test statistic < z

< t

: H0 will not be rejected if z-test is used.

9 CommonAssumptions

1. For t-test, we may need to assume population is normallydistributed if sample size is given to be small and population

variance is unknown.

2. For z-test, we do not need to assume population is normally

distributed if sample size is large.3. Sample are independently chosen.

hypothesis testing skills set

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