Test Of Hypothesis
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Test of Hypothesis H1: μ = 368 H2: μ ≠ 368 1. Null hypothesis H0 is the hypothesis that is always tested. 2.
The alternative hypothesis H1, is the opposite of the Null Hypothesis and represents the conclusion supported if the Null Hypothesis is rejected.
3.
The Null Hypothesis H0 always refers to a specified value of the population parameter (such as μ), not a sample statistic (such as X).
4.
The statement of the Null Hypothesis always contains an equal sign regarding the specific value of the population parameter (e.g. H0: μ = 368 gms)
5.
Statement of Alternative Hypothesis never contains an equal sign regarding the specified value of the population parameter u ≠368 gms.
Reasons for Rejection & Non Rejection Level of Significance: α = 0.1,0.5,0.1 Complement (1- α) = Confidence Coefficient Type 1 Error = Level of Significance Occurs if the Null Hypothesis H0 is rejected when in fact it is true and should not be true. The probability of Type 1 error occurring is α. Type II error = Occurs if the Null Hypothesis H0 is not rejected when in fact it is false and should be rejected. Type II error is β. Unlike Type I error which is controlled by selection of α, the probability of making Type II error is dependent on the difference between hypothetical and actual values. If the difference is larger, the chance of committing Type II error is less. COMPLEMENT OF PROBABILITY OF TYPE II ERROR IS (1- β) IS THE POWER OF A STATISTIC.
10 Step Method of Hypothesis Testing: 1. State the Null Hypothesis H0 2.
State the Alternative Hypothesis H1
3.
Choose the level of significance α
4. Choose the sample size n 5. Conduct Z test 6. Set up Critical Values that divide the Rejection and Non Rejection region. 7. Determine whether sample statistics fall into the regions. 8.
Make the Statistical decision.
9.
Express the decision in context of the problem: There is conclusive evidence that the average amount filled is different from 368 gms.
Connection between Confidence Level Estimation and Hypothesis Testing:
n=25, X=372.5, σ=15 From confidence level of 95% (Corresponding to 0.05 level of significance)
Statistical Decision Do not reject H0 Reject H0
H0 (True)
H0 (False)
Correct decision confidence 1- α Type I error P = α
Type II error P = μ Correct decision Power = 1- μ
366.2 ≤ μ ≤ 378.38
Because the interval includes the hypothetical value of 368 gms, the Null Hypothesis is not rejected. There is insufficient evidence that the Mean Fill Amount over the entire filling process is not 368 gms. This is the same decision as in Hypothesis Testing.
Actual Situation
TYPE II error can be reduced by increasing the size of the sample. Risks of Type I and Type II error determines values of α and β. Reduction in Type I error results in increase of Type II error.
TWO TAIL Z TEST:
If level of significance is 0.05, size of rejection region is . 05 & critical values of Normal Distribution can be determined.
-1.96
368
Critical Value
+1.96 Critical Value
Decision Rule: Reject H0 if Z > 1.96 or if Z< 1.96 Otherwise do not reject H0 Suppose the sample of 25 cereal boxes indicates a sample mean X = 372.5 & population standard deviation σ is assumed to be 15 gms
Because Z = +1.50 and -1.96 < 1.50 < +1.96 Decision not to reject H0 Alternatively, to take into account the possibility of a Type II error, the conclusion can be stated as “there is insufficient evidence that the mean fill is different from 368 gms”. Beautiful
The alternative hypothesis H1, is the opposite of the Null Hypothesis and represents the conclusion supported if the Null Hypothesis is rejected.
3.
The Null Hypothesis H0 always refers to a specified value of the population parameter (such as μ), not a sample statistic (such as X).
4.
The statement of the Null Hypothesis always contains an equal sign regarding the specific value of the population parameter (e.g. H0: μ = 368 gms)
5.
Statement of Alternative Hypothesis never contains an equal sign regarding the specified value of the population parameter u ≠368 gms.
Reasons for Rejection & Non Rejection Level of Significance: α = 0.1,0.5,0.1 Complement (1- α) = Confidence Coefficient Type 1 Error = Level of Significance Occurs if the Null Hypothesis H0 is rejected when in fact it is true and should not be true. The probability of Type 1 error occurring is α. Type II error = Occurs if the Null Hypothesis H0 is not rejected when in fact it is false and should be rejected. Type II error is β. Unlike Type I error which is controlled by selection of α, the probability of making Type II error is dependent on the difference between hypothetical and actual values. If the difference is larger, the chance of committing Type II error is less. COMPLEMENT OF PROBABILITY OF TYPE II ERROR IS (1- β) IS THE POWER OF A STATISTIC.
10 Step Method of Hypothesis Testing: 1. State the Null Hypothesis H0 2.
State the Alternative Hypothesis H1
3.
Choose the level of significance α
4. Choose the sample size n 5. Conduct Z test 6. Set up Critical Values that divide the Rejection and Non Rejection region. 7. Determine whether sample statistics fall into the regions. 8.
Make the Statistical decision.
9.
Express the decision in context of the problem: There is conclusive evidence that the average amount filled is different from 368 gms.
Connection between Confidence Level Estimation and Hypothesis Testing:
n=25, X=372.5, σ=15 From confidence level of 95% (Corresponding to 0.05 level of significance)
Statistical Decision Do not reject H0 Reject H0
H0 (True)
H0 (False)
Correct decision confidence 1- α Type I error P = α
Type II error P = μ Correct decision Power = 1- μ
366.2 ≤ μ ≤ 378.38
Because the interval includes the hypothetical value of 368 gms, the Null Hypothesis is not rejected. There is insufficient evidence that the Mean Fill Amount over the entire filling process is not 368 gms. This is the same decision as in Hypothesis Testing.
Actual Situation
TYPE II error can be reduced by increasing the size of the sample. Risks of Type I and Type II error determines values of α and β. Reduction in Type I error results in increase of Type II error.
TWO TAIL Z TEST:
If level of significance is 0.05, size of rejection region is . 05 & critical values of Normal Distribution can be determined.
-1.96
368
Critical Value
+1.96 Critical Value
Decision Rule: Reject H0 if Z > 1.96 or if Z< 1.96 Otherwise do not reject H0 Suppose the sample of 25 cereal boxes indicates a sample mean X = 372.5 & population standard deviation σ is assumed to be 15 gms
Because Z = +1.50 and -1.96 < 1.50 < +1.96 Decision not to reject H0 Alternatively, to take into account the possibility of a Type II error, the conclusion can be stated as “there is insufficient evidence that the mean fill is different from 368 gms”. Beautiful
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