Miskolci Egyetem Gazdaságtudományi Kar Üzleti Információgazdálkodási és Módszertani Intézet. Hypothesis Testing. Petra Petrovics

Miskolci Egyetem Gazdaságtudományi Kar Üzleti Információgazdálkodási és Módszertani Intézet

Hypothesis Testing

Petra Petrovics PhD Student


Inference from the Sample to the Population Estimation

Hypothesis Testing

Estimation: how can we determine the value of an unknown parameter of a population by using the sample. Hypothesis testing: how to test a statement concerning a population parameter.


Hypotheses • Null hypothesis H0 The hypothesis to be tested. A proposition that is considered true unless the sample we use to conduct the hypothesis test gives convincing evidence that the null hypothesis is false.

• Alternative hypothesis H1 It is the hypothesis accepted when the null hypothesis is rejected.


Hypothesis Testing • Null hypothesis: H0 :  = 0 • Alternative hypothesis: H1 :   0 Two-tailed test   0 One-tailed test   0


Basic Terms I • Statistical test A decision function that takes its values in the set of hypotheses. • Region of acceptance The set of values for which we fail to reject the null hypothesis. • Region of rejection / Critical region The set of values of the test statistic for which the null hypothesis is rejected.


Basic Terms II • p-value The probability, assuming the null hypothesis is true, of observing a result at least as extreme as the test statistic. • Significance level: α → a value indicating the percentage of sample values that is outside certain limits → the probability of rejecting the null hypothesis when it is true • Critical values: limits of the rejection region


Actual situation Conclusion

Fail to reject H0

Reject H0

H0 true

H0 false

Correct decision 1-

Type I error 

Type II error

Correct decision


The Five-Step Procedure for Hypothesis Testing 1. Set up the null hypothesis H0 and the alternative hypothesis H1. 2. Define the test statistic. The test statistic will be evaluated, using the sample data, to determine if the data are compatible with the null hypothesis. 3. Define a rejection region, having determined a value for α. 4. Carry out the test. State our decision: to reject H0 or to fail to reject H0. 5. Give a conclusion! (In the terms of the original problem or question)


Hypothesis Testing for the Population Mean H0 : μ = m0 1.) Population with normal distribution, σ known z =

x - m0 σ/ n

2.) Population with normal distribution, σ unknown, n  100 x - m0 t= s/ n x - m0 3.) σ unknown, n  100 z= s/ n


• Critical Values in case of Large Sample

zπ

H1 :   m0

H1 :   m0

Pr

0

 z1

z1 / 2

H1 :   m0

Pr

Pr

0

0

z1

z1 / 2

• Critical Values in case of Small Sample

tπ

H1 :   m0

t1

H1 :   m0

Pr

0

t1 / 2

H1 :   m0

Pr

Pr

0

t1 / 2

0

t1


Hypothesis Testing for a Population Proportion H0 : P = P0 In case of a sample when n ≥ 100 ! z

p - P0 P0Q0 /n

Hypothesis Testing for a Population Standard Deviation H0 : σ = σ0 Only in the case when the population distribution is 2 normal. 2 n - 1  s  

σ 02


• Critical values of χ2-test

χ2

H1 :    0

Pr

0

2

H1 :    0

Pr

2

0

2 / 2

Pr

12 / 2

2

H1 :    0

0

12

2


Exercise The data below are a random sample of a whole coffee filling machine which fills 250 g bags of coffee. The distribution of the weight in the population is normal. Weights of the bags (g)

Number of bags

– 240

8

241 – 245

22

246 – 250

32

251 – 255

28

255 –

10

Total

100


a) Estimate the average weight of the coffee bags! Construct a 95 percent confidence interval for the population mean! b) The management is actually saying that it wants a more precise estimation. What sample size will provide a 20% less maximum error? c) Construct a 95% confidence interval for the population standard deviation! d) How much coffee does this machine need to fill in the bags if it produces 10 000 coffee bags per day? (π=95%) e) Estimate the proportion of the bags in the population which contain not more than 250 g coffee ( = 99 %)! f) Do the data support the statement that the mean of the population is 250 g ? ( = 1 %) g) Test the hypothesis that the population standard deviation is 6g! Use a 5% significance level! h) Test the hypothesis that the proportion of the coffee bags containing more than 250g coffee is at least 40% !


Comparing Two Population Means H0 : μ1 – μ2 = δ Sample size Data

Sample 1 m

Sample 2 n

x11, x12, ..., x1m

x21, x22, ..., x2n

Sample mean

x1

x2

Sample variance

s12

s 22

a) Using two small independent samples and normal populations

b) Using two large independent samples

d-

t=

Assumption: equal standard deviations z=

d- sd

sp

1 1  n m


Comparing Two Population Proportions H0 : P1 – P2 = ε

Sample size Proportion Standard deviation where

Sample 1 m

Sample 2 n

p1  k1 / m

p2  k 2 / n

s1 

s2 

p1q1

q1 = 1 - p1

Using large independent samples

p2 q2

q2 = 1 - p2

e z se


Comparing the Variances of Two Normal Populations H0 :  1 =  2 Using independent samples: H1 H1: 1≠2

Probability 1-/2

H1: 1<2

1-

H1: 1>2

1-

Lower critical value (cl) F

1



1 ( 2 ; 1 )

2

1 F1 ( 2 ; 1 )

0

F=

s12 s 22

Upper critical value (cu) F

1



( 1 ; 2 )

2

-

F1 ( 1 ; 2 )


• Critical values of F-test

F

Pr

Pr

0

HPr1 : H1 :2   Pr H1 1:  1 1  2 2

1 1 0 F1 (02 ; 1 ) 1 F1 ( 2 ; 1 ) F1 ( 2 ; 1 )

HPr1 : H1 :2   H1 :1 1 1  2 2

Pr

1 1 0 F ( F 1 0  2 ;F1 )1 ( 2 ; 1 ) 2 F F  1( 2 ; 1 ) 0

F

1

2

Pr

Pr

F

1

 ( 1 ; 2 ) F  ( 1 ; 2 ) 2 F  (112; 2 ) 1 2

F

F 0

HPr1 : H 1 2 H1 : 11:  12  2

F 0

0

F1 ( 1 ; 2 ) F1 ( 1 ; 2 ) F1 ( 1 ; 2 )

F

F


Exercise 6 (cont.) i) The management wants to buy a new filling automat which fills the coffee bags more precisely. Test the statement that the standard deviation of weight of bags produced by the new machine is not equal to the standard deviation of the old machine. A sample of 150 coffee bags was taken and their total weight was 37.65kg (x2= 9 454 322). α = 0.01 j) Should the management conclude, at the 2 percent significance level, that the new machine fills at least 7 g more coffee per bags?


Exercise 7 The data below are a random sample of a tube cutting machine which cuts 1200 mm long pieces of tubes.

1208; 1204; 1202; 1202; 1194; 1195; 1205; 1194; 1197; 1193; 1205; 1202; 1191; 1195; 1194 a) Test the hypothesis that the population standard deviation is 3 mm! Use a 5% significance level! b) Do the data support the statement that the mean of the population is 1200 mm? (α=5%)


Exercise 8 Doctors wants to compare the heigh of basketball players and swimmers. The following sample of 14 basketball player was taken (in cm): 198; 202; 199; 202; 191; 198; 199; 205; 204; 200; 199; 199; 200; 204 The average heigh of the 10 swimmers is 196 cm, its standard deviation is 5.2 cm. (The distribution of the heigh is normal.) a) Determine a 98% confidence interval for the average heigh of basketball players. b) At what significance level can you accept that at least half of the basketball players is higher than 2m? c) Test the hypothesis that the average heigh of basketball players is more than that of the swimmers and the difference is not more than 5 cm.


Exercise 9 From the 16,200 female employees of a multinational company a sample of 162 was taken. Their total age is 5,303 years (Σx2=188110). a) Estimate the average age (π=98.8%). b) What sample size will provide half of the maximum error? c) In the sample there are 112 women younger than 35. Would you accept the statement that 75% of the women is younger than 35? (α=5%) d) Another sample was drawn from men. The average age of the 121 men is 36.9. (s=9.504). At what significance level can you accept that the average age of the women is less than that of the man by 3 years?


Exercise 10 • To increase the amount of the trade, McDonald’s tried two kinds of promotions. On Monday it reduced the price of the menu, and on Tuesday free ice was given with the menu. To test which promotion is more successful the trade was examined in 10 McDonald’s: 1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

Monday

162

102

109

98

121

105

139

106

99

93

Tuesday

153

88

122

93

112

108

135

97

104

85

• Test the statement that price reduction was more successful. (α = 5%)


Thank You for Your Attention!

Miskolci Egyetem Gazdaságtudományi Kar Üzleti Információgazdálkodási és Módszertani Intézet. Hypothesis Testing. Petra Petrovics

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