Unit 7 – Hypothesis Testing Practice Problems SOLUTIONS

[Pages:9]PubHlth 540 ? Fall 2011

Introductory Biostatistics

Unit 7 ? Hypothesis Testing

Practice Problems SOLUTIONS

Page 1 of 9

1. An independent testing agency was hired prior to the November 2010 election to study whether or not the work output is different for construction workers employed by the state and receiving prevailing wages versus construction workers in the private sector who are paid rates determined by the free market. A sample of 100 private sector workers reveals an average output of 74.3 parts per hour with a sample standard deviation of 16 parts per hour. A sample of 100 state workers reveals an average output of 69.7 parts per hour with a sample standard deviation of 18 parts per hour. In developing your answer, you may assume that the unknown variances are equal.

(a) Is there evidence of a difference in productivity at the 0.10 level of significance? (b) Is there evidence of a difference in productivity at the 0.05 level of significance? (c) What is the achieved level of significance?

ANSWER

a. The p-value is less than 0.1, so it is significant at 0.1 level. b. The p-value is bigger than 0.05, so it is not significant at 0.05 level. c. The achieved level is 0.058.

SOLUTION This question is asking for a hypothesis test of the equality of two means in the setting of two independent groups (state v private) . Research Question. Is the work output of state workers is different from that of workers in the private sector?

Assumptions. Let subscript "1" reference the group of state employees, "2" the private sector employees. X1 is distributed Normal (1 , 2/100) and X2 is distributed Normal (2 , 2/100)

HO and HA.

HO : 1 = 2 HA : 1 2

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PubHlth 540 ? Fall 2011

Introductory Biostatistics

Page 2 of 9

Test statistic is a t-score.

t

score

=

(X1

-X2 ) - E[(X1 SE^ [(X1-X2 )

-X2 | H

)

O

| HOtrue] true]

It's okay to assume equality of unknown variances (because I said it was

( ) SE^ X1-X2

=

S2 pool

+

S2 pool

n1 n2

where

S2 pool

=

(n1-1)S12 +(n2 -1)S22

(n1-1)+ (n2 -1)

For these data:

^ 2 =S2pool =

(n1-1)S12 + (n2 -1)S22 (n1-1) +(n2 -1)

=

(100-1)182 + (100-1)162 (100-1) +(100-1)

=290

( ) SE^ X1-X2

=

S2 pool

+

S2 pool

=

n1 n2

290 + 290 =2.4083 100 100

Degrees of freedom = (n1-1) + (n2-1) = (100-1) + (100-1) = 198. "Evaluation" rule.

The likelihood of these findings or ones more extreme if HO is true is

( ) p-value = Pr X1-X2 |(69.7-74.3)|HOtrue .

Calculations.

( ) p-value = (2)Pr X1-X2 |(69.7-74.3)| note ? The (2) is in front because this is two sided

=

2

Pr

(

X1 SE^

- X2) ( X1 -

-(0)

X2)

|

(

69.7

- 74.3)

2.4083

-

(0)

|

= [ (2)Pr tscore 1.91] where degrees of freedom = 198

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PubHlth 540 ? Fall 2011

Introductory Biostatistics

Page 3 of 9

=(2)(.028)=.056 note ? I used the Normal(0,1) table as degrees of freedom is so large

"Evaluate".

Under the null hypothesis HO (worker output is the same in both groups) the chances that the average work outputs differ by a magnitude greater than | 69.7 ? 74.3 | is about 6 in 100. This is a borderline suggestion that the two groups differ in their work output.

2. For the data in Exercise 1, what level of significance is achieved by the data if the sample means and sample standard deviations are unchanged but the within group sample sizes are

(a) both equal to 10 (b) both equal to 200 (c) Comment on the role of sample size in the probability of a type I error.

ANSWER

a. p-value = .554 b. p-value = .007 c. All other things equal, a larger sample size reduces type I error.

SOLUTION The solution involves substitution of the new values of the sample sizes into the formulae shown in the solution for Exercise 1.

a. n=10 in each group

^ 2 =Sp2ool

=

(n1-1)S12 + (n2 -1)S22 (n1-1) + (n2 -1)

=

(10-1)182 + (10-1)162 (10-1) +(10-1)

=290

( ) SE^ X1-X2

=

S2 pool

+

S2 pool

=

n1 n2

290 290 + =7.6158

10 10

Degrees of freedom = (n1-1) + (n2-1) = (10-1) + (10-1) = 18.

( ) p-value = (2)Pr X1-X2 |(69.7-74.3)| note ? The (2) is in front because this is two sided

=

2

Pr

(

X1 SE^

- X2) ( X1 -

-(0)

X2)

|

(69.7

- 74.3)

7.6158

-

(

0)

|

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PubHlth 540 ? Fall 2011

Introductory Biostatistics

Page 4 of 9

= [ (2)Pr tscore 0.6040] where degrees of freedom = 18

= (2) (.2767) = .55

b. n=200 in each group

^ 2 =Sp2ool =

(n1-1)S12 + (n2 -1)S22 (n1-1) +(n2 -1)

=

(200-1)182 + (200-1)162 (200-1) +(200-1)

=290

( ) SE^ X1-X2

=

S2 pool

+

S2 pool

=

n1 n2

290 + 290 =1.7029 200 200

Degrees of freedom = (n1-1) + (n2-1) = (200-1) + (200-1) = 398.

( ) p-value = (2)Pr X1-X2 |(69.7-74.3)| note ? The (2) is in front because this is two sided

=

2

Pr

(

X1 SE^

- X2) ( X1 -

-(0)

X2)

|

(69.7

- 74.3)

1.7029

-

(

0)

|

= [ (2)Pr tscore 2.7013] where degrees of freedom = 398

= (2) (.0036) = .0072

3. Halcion is a sleeping pill that is relatively rapidly metabolized by the body and therefore having fewer hangover effects the next morning, compared to other sleeping pills. Opponents of Halcion argue that, because this agent is so rapidly metabolized by the body, patients do not sleep as long with this drug as with Dalmane. Data on 10 insomniacs, each of whom took Dalmane on one occasion and Halcion on a second, is collected. The variable measured is number of hours of sleep:

Patient 1 2 3 4 5 6 7 8 9 10

Number of Hours Sleep with

Dalmane

Halcion

4.58

3.97

5.19

4.88

3.94

4.09

6.32

5.87

7.68

6.93

3.48

4.00

5.72

5.08

7.04

6.95

5.27

4.96

5.84

5.13

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PubHlth 540 ? Fall 2011

Introductory Biostatistics

Page 5 of 9

Do these data suggest that Halcion is not as effective as Dalmane with respect to number of hours of sleep? Carry out an appropriate statistical test and interpret your findings.

ANSWER

Yes, a paired t-test suggests that the average difference in hours slept (Dalmane ? Halcion) = 0.32 is statistically significant (one sided p-value = .018).

SOLUTION This question is asking for a hypothesis test of the equality of two means in the setting of paired data. The data are paired because each participant was measured on two occasions, once on Dalmane and once on Halcion . Research Question. Are sleep durations shorter on Dalmane than on Halcion?

Assumptions. d is distributed Normal (d , d2/10) Differences are calculated as (Dalmane ? Halcion)

For these 10 paired measurements, we have

Obs dalmane halcion

diff

1

4.58

2

5.19

3

3.94

4

6.32

5

7.68

6

3.48

7

5.72

8

7.04

9

5.27

10

5.84

3.97 4.88 4.09 5.87 6.93 4.00 5.08 6.95 4.96 5.13

0.61 0.31 -0.15 0.45 0.75 -0.52 0.64 0.09 0.31 0.71

HO and HA.

HO : d = 0 HA : d > 0 ("Dalmane is better than Halcion) ? one sided

sol_testing.doc

PubHlth 540 ? Fall 2011

Introductory Biostatistics

Test statistic is a t-score.

t

score

=

(d)-E[d)|H O true] SE^ [(d)|HOtrue]

Obtain sample mean of the differences, d

10

di

d = i=1 = 0.32 10

Preliminary ? Obtain sample variance of the differences, Sd2

( ) 10 di - d 2

10

(di - 0.32)2

Sd2 =

i=1

(n-1)

= i=1 9

=0.1688889

Obtain SE^ [ d | HO true]

SE^ [ d | HO true] =

Sd2 = n

0.1688889 = 0.1299573 10

Putting these all together, the solution for the test statistic is

t score

=

(d)-E[d)|H O true] SE^ [(d)|HOtrue]

=

0.32 - 0 0.1299573

=

2.4623

Degrees of freedom = (n-1) = (10-1) = 9.

"Evaluation" rule. The likelihood of these findings or ones more extreme if HO is true is

( ) p-value = Pr d 0.32 | HO true .

Page 6 of 9

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PubHlth 540 ? Fall 2011

Introductory Biostatistics

Page 7 of 9

Calculations.

p-value = [Pr tscore 2.46] where degrees of freedom = 9

=.018 If you want to use a student's t-distribution calculator on the internet, one choice is



Enter the following, being sure to click on the radio dial for a RIGHT TAIL

After pressing the RIGHT ARROW, you should obtain 0.0181 in the probability box.

"Evaluate". Under the null hypothesis HO (duration of sleep is the same with both drugs) the chance that the

sol_testing.doc

PubHlth 540 ? Fall 2011

Introductory Biostatistics

Page 8 of 9

difference in average hours slept is as great or greater than 0.32 hours is about 2 in 100. This is statistically significant.

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