![57
Hypothesis about μ, σ known
Example 1
A machine producing plastic containers is
properly adjusted if the average thickness of
plastic is 11 mm. (population standard deviation is
known to be 0.18)
A random sample of 64 containers had an average
thickness of 10.94 mm
Is the machine properly adjusted?
[n=64; x̄ = 10.94; σ=0.18]](https://image.slidesharecdn.com/brmtopic9hypothesistestingmsk-221112135624-78ca0eae/85/Hypothesis-testing-57-320.jpg)
![59
3) Test statistic
Remember
If large sample (n≥30) then
the population of sample means x̄ is
approximately N(μx̄ , σx̄ ) & μx̄ = μ , σx̄= σ/√n
Transformation to standardise
Test statistic
[ µ not known but use the value of 11 from H0 ]
n
x
n
x
z
11
−
=
−
=](https://image.slidesharecdn.com/brmtopic9hypothesistestingmsk-221112135624-78ca0eae/85/Hypothesis-testing-59-320.jpg)
![63
Hypothesis about μ, σ unknown
Example 2
Monthly returns normally distributed, industry mean
return 3.6%;
Sample taken, monthly returns :-1.3; 2.7; 4.1; 2.5
Are the returns consistent with the industry mean of
3.6%? [ Test if μ = 3.6 ]
Solution
1) H0 : μ = 3.6
Ha : μ ≠ 3.6
2) α = 0.10](https://image.slidesharecdn.com/brmtopic9hypothesistestingmsk-221112135624-78ca0eae/85/Hypothesis-testing-63-320.jpg)
![64
3) Test statistic (σ unknown)
3) Sample statistic:
[ unknown µ replaced with the value from H0 , ‘for
the time being’ ]
n
s
x
n
s
x
t
6
.
3
−
=
−
=
](https://image.slidesharecdn.com/brmtopic9hypothesistestingmsk-221112135624-78ca0eae/85/Hypothesis-testing-64-320.jpg)
![68
Hypothesis about proportion
The population proportion is the number of
members in the population with a particular attribute
divided by the number of members in the population.
For large samples
Population proportion π
Sample proportion
Test statistic
[π not known but use the value from H0 ]
( )
n
p
z
−
−
=
1
ˆ
n
x
p =
ˆ](https://image.slidesharecdn.com/brmtopic9hypothesistestingmsk-221112135624-78ca0eae/85/Hypothesis-testing-68-320.jpg)
Hypothesis testing
- 1. Chapter 21 Hypothesis Testing Business Research Methodology
- 2. Normal Distribution A normal distribution, sometimes called the bell curve, is a symmetrical distribution that occurs naturally in many situations. For example, the bell curve is seen in tests like the SAT and GRE. The bulk of students will score the average (C), while smaller numbers of students will score a B or D. An even smaller percentage of students score an F or an A.
- 3. Properties of a normal distribution 1. The mean, mode and median are all equal. 2. The curve is symmetric at the center (i.e. around the mean, μ). 3. Exactly half of the values are to the left of center and exactly half the values are to the right. 4. The total area under the curve is 1. 5. A standard normal distribution has a mean of 0 and a standard deviation of 1.
- 8. Z-Value ▪ For Two-tailed test, Z-value for 90%, 95%, and 99% confidence interval are 1.645, 1.96, and 2.58 respectively. ▪ For One-tailed test, Z-value for 90%, 95%, and 99% confidence interval are 1.28, 1.645, and 2.33 respectively. ▪ For Two-tailed test, Z-value for 10%, 5%, and 1% significance level (α) are 1.645, 1.96, and 2.58 respectively. ▪ For One-tailed test, Z-value for 10%, 5%, and 1% significance level (α) are 1.28, 1.645, and 2.33 respectively.
- 10. What is a Hypothesis? • A hypothesis is an unproven proposition or assumption about the relation between two or more variables that is to be tested based on empirical data. – Ex: There is a positive relation between sales and profit. – Ex: There is no relation between weather change and budget deficit • This assumption may or may not be true. • Hypothesis testing refers to the formal procedures used by statisticians to accept or reject statistical hypotheses.
- 11. 2 Types of Hypothesis: Null vs. Alternative
- 12. 2 Types of Hypothesis: Null vs. Alternative (cont’d)
- 13. 13 Correct decision or error? Type I vs. Type II error Null Hypothesis (in reality): What to do Do not reject H0 Reject H0 H0 is true Correct decision Type I error (α risk) H0 is not true Type II error (β risk) Correct decision
- 14. 14 Correct decision or error? Type I vs. Type II error (cont’d) H0: The patient has NO relation with pregnancy.
- 22. Z-test vs t-test ▪ Z-test is used when sample size is large (n>50), or the population variance is known. ▪ t-test is used when sample size is small (n<50) and population variance is unknown. ▪ If sample size>30, the t-test procedure gives almost identical p-values as the Z-test procedure
- 32. Stata,
- 55. Type in Stata command box the following after the regression is done: test SMB=HML=MOM=0
- 56. Problem on Hypothesis Testing 56
- 57. 57 Hypothesis about μ, σ known Example 1 A machine producing plastic containers is properly adjusted if the average thickness of plastic is 11 mm. (population standard deviation is known to be 0.18) A random sample of 64 containers had an average thickness of 10.94 mm Is the machine properly adjusted? [n=64; x̄ = 10.94; σ=0.18]
- 58. 58 Example 1 - Solution 1) State H0 and Ha H0 : machine properly adjusted H0 : μ = 11 Ha : machine is not properly adjusted Ha : μ ≠ 11 2) Use α = 0.05 = 5%
- 59. 59 3) Test statistic Remember If large sample (n≥30) then the population of sample means x̄ is approximately N(μx̄ , σx̄ ) & μx̄ = μ , σx̄= σ/√n Transformation to standardise Test statistic [ µ not known but use the value of 11 from H0 ] n x n x z 11 − = − =
- 60. 60 4) Decision Rule We don’t expect precisely 11 mm but something close enough (to 11 mm/ to 0). ‘Close enough’ depends on α =0.05 (1- α) =0.95 0.95 α/2=0.05/ 2=0.025 α/ 2 z0 -z0 z0 = 1.96 0
- 61. 61 4) Decision rule cont. Reject H0 if zcalculated >1.96 or zcalculated < -1.96 Do not reject H0 Rejection region Rejection region z0 -z0 z0 = 1.96
- 62. 62 5) Sample examined Calculate test statistic zcalc < -1.96, so Reject H0 (conclude that machine not properly adjusted) Business decision: What to do? New machine, old machine to fix, do nothing 5% chance of Type I error 667 . 2 64 18 . 0 11 94 . 10 11 − = − = − = n x zcalc
- 63. 63 Hypothesis about μ, σ unknown Example 2 Monthly returns normally distributed, industry mean return 3.6%; Sample taken, monthly returns :-1.3; 2.7; 4.1; 2.5 Are the returns consistent with the industry mean of 3.6%? [ Test if μ = 3.6 ] Solution 1) H0 : μ = 3.6 Ha : μ ≠ 3.6 2) α = 0.10
- 64. 64 3) Test statistic (σ unknown) 3) Sample statistic: [ unknown µ replaced with the value from H0 , ‘for the time being’ ] n s x n s x t 6 . 3 − = − =
- 65. 65 4) Decision rule – stats tables required For critical value for (4): df = n-1 = 3 & α = 0.1=10% From tables : Do not reject H0 Rejection region = 0.1/2 = 0.05 Rejection region = 0.1/2 = 0.05 t0 -t0 t0 = 2.353 Acceptance region
- 66. 66 4) Decision rule Reject H0 if tcalculated >2.353 or tcalculated < -2.353 Do not reject H0 Rejection region Rejection region t0 -t0 t0 = 2.353 Acceptance region
- 67. 67 5) Sample examined Do not reject H0 ( ) 39 . 1 4 31 . 2 6 . 3 2 6 . 3 31 . 2 35 . 5 35 . 5 1 0 . 2 4 5 . 2 1 . 4 7 . 2 3 . 1 2 2 − = − = − = = = = − − = = + + + − = = n s x t s n x x s n x x calculated i i
- 68. 68 Hypothesis about proportion The population proportion is the number of members in the population with a particular attribute divided by the number of members in the population. For large samples Population proportion π Sample proportion Test statistic [π not known but use the value from H0 ] ( ) n p z − − = 1 ˆ n x p = ˆ
- 69. 69 Hypothesis about proportion, also two-sided and one-sided tests Example 3 Manufacturer claims that no more than 20% of the parts are defective. Test at 1% significance level. A sample of 120 taken and it has 32 defective items. Solution 1) H0 : π = 0.20 (Manufacturer claims so) Ha : π > 0.20 (That’s what we think/suspect) 2) α = 0.01 3) Test statistic (as stated before)
- 70. 70 Example 3 - Solution 4) Decision rule 5) Do not reject H0 0 2.3263 Rejection region α = 0.01 Acceptance region Reject H0 if Zcalc > 2.3263 z scale 826 . 1 120 80 . 0 * 20 . 0 20 . 0 267 . 0 80 . 0 * 20 . 0 20 . 0 267 . 0 120 32 = − = − = = = n p z p calc
- 71. 71 Hypothesis about variance Example 4 The variance of return on a ‘moderately safe’ portfolio should not exceed 15%. For a sample of 10 stocks the standard deviation was 6.5% (s=6.5). What can be concluded at 5% significance level? Solution 1) H0 : σ2 = 15 Ha : σ2 > 15 2) α = 0.05
- 72. 72 Test statistic and decision rule χ2 χ 2 5%, 9 = 16.919 Rejection area Do not reject H0 ( ) 2 2 2 1 s n − =
- 73. 73 Example 4: Solution Calculated test statistic: From tables: χ 2 5%, 9 = 16.919 Conclusion: Reject H0 , (conclude that portfolio seems riskier) ( ) ( ) 35 . 25 15 5 . 6 1 10 1 2 2 2 2 = − = − = s n
- 74. 74 Hypothesis about two population means (independent samples, population variances known) Example 5 Population 1 (ABC Shipping), Sample 1: n1 = 45, x̄1= 16, (σ1 = 3.5) Population 2 (Speedy Air Freight), Sample 2: n2 = 55, x̄2= 14, (σ2 = 6.2) Can it be concluded that Speedy significantly quicker?
- 75. 75 Example 5: H0 and H a H0 : 1 = 2 ; Ha : 1 > 2 (=Speedy delivers quicker) (1 - 2 > 0) (What do we hope to show?) One sided test needed
- 76. 76 Different critical values depending on α H0 : 1 = 2 ; Ha : 1 > 2 (=Speedy delivers quicker) (1 - 2 > 0) Reject H0 zta b Different ztab , depending on α Reject H0 if zcalculated > ztabulated
- 77. 77 See below test statistic for inference about two populations Test statistic and calculations ( ) ( ) ( ) ( ) 0295 . 2 55 2 . 6 45 5 . 3 0 14 16 0 2 2 2 2 2 1 2 1 2 1 2 2 2 1 2 1 2 1 2 1 = + − − = + − − = + − − − = n n x x n n x x zcalculated
- 78. 78 Two Populations Zcalculated = 2.0295 If α =10%, ztab = 1.2816, Reject H0 If α =5%, ztab = 1.6449 Reject H0 If α =1%, ztab = 2.3263, Do not reject H0 Reject H0 Different ztab , depending on α Reject H0 if zcalculated > ztabulated zta b
- 79. 79 Different α – different conclusions Rejection tail 2.3263 Reject H0 , conclude that Speedy better BUT 5% chance that we rejected even though H0 true α = 5% 1.6449 α = 1% Do not reject H0 = keep believing that ABC & Speedy equally fast Zcalculated = 2.0295
- 80. 80 Useful formulas ( ) ( ) ( ) ( ) 2 2 2 1 2 1 2 1 2 1 2 2 2 1 1 ˆ n n x x z s n n p z n s x t n x z + − − − = − = − − = − = − =
- 81. Thank You!!! 81