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STATISTICAL HYPOTHESIS
A Statistical Hypothesis is an assumption about an unknown population parameter. A hypothesis can be of two types:
1. Simple Hypothesis If the values of the parameters under a given hypothesis are specified, it is called Simple Hypothesis. 2. Composite Hypothesis If any of the values of the parameter is not completely specified under the given hypothesis, it is called a Composite Hypothesis.
Hypothesis Testing The process of making a decision on whether to accept or reject an assumption about a population parameter on the basis of sample information is called Hypothesis Testing.
HYPOTHESIS TESTING PROCEDURE
Step 1: State the Null Hypothesis (H 0 ) and Alternative Hypothesis (H 1 )
Null Hypothesis It is the assumption or hypothesized parameter value which is tested for possible rejection or acceptance on the basis of sample information. It is always expressed in the form of an equation that assigns a specific value to the population parameter. Thus, H 0 : μ = 5. Alternative Hypothesis It is the logical opposite of the null hypothesis. This implies, if null hypothesis is found to be false, the alternative hypothesis must be true and vice versa. The Alternative Hypothesis states that a specific population parameter is not equal to the value stated in Null Hypothesis. Thus, H 1 : μ 5 Therefore: H 1 : μ < 5 or H 1 : μ > 5
Step 2: State the Level of Significance (α) for the test
Level of significance is the maximum probability that the null hypothesis will be rejected when it is true. It is usually expressed in % and is denoted by α. A 5% level of significance implies that there are 5 out of 100 chances that a null hypothesis is rejected when it is true. Thus, it means that there is a 5% chance of making a wrong decision or it can be inferred as 95% confidence level that correct decision is made.
Step 3: Establishing critical or rejection region
Test Statistic The statistic on which the test procedure is based is called the Test Statistic. The decision to accept or reject the null hypothesis is made on the basis of the test statistic computed from the sample observations. The test statistic should be the one whose sampling distribution is known under the assumption that the null hypothesis is true.
Acceptance and Rejection Regions The area under the sampling distribution curve of the test statistic is divided into two mutually exclusive regions called the Acceptance Regions and the Rejection or Critical Region. If the value of the test statistic falls into the acceptance region, the null hypothesis is accepted otherwise it is rejected. The size of the critical region is directly related to the level of significance. The value of sample statistic that separates the regions of acceptance and rejection is called the critical value. There are three types of test on the basis of the way null and alternative hypothesis are formed.
Two-tailed Tests For a two-tailed or Double-tailed test, the null hypothesis and alternative hypothesis are stated as under: H 0 : μ = μ 0 and H 1 : μ ≠ μ (^0) Following figure shows a two-tailed test:
Right-tailed Test The hypothesis for a right-tailed test is usually expressed as: H 0 : μ ≤ μ 0 and H 1 : μ > μ (^0) Following figure shows a right-tailed test:
Left-tailed Test The hypothesis for a left-tailed test is usually expressed as: H 0 : μ ≥ μ 0 and H 1 : μ < μ (^0) Following figure shows a left-tailed test:
Step 4 : Calculation of Suitable Test Statistic The value of test statistic is calculated from the distribution of sample statistic by the following formula:
The choice of a probability distribution is guided by the sample size and the value of the population standard deviation as shown below:
Sample Size Population Standard Deviation Known Unknown Large Sample n > 30 Normal Distribution Normal Distribution Small Sample n ≤ 30 Normal Distribution t -distribution
Step 5: Drawing a Conclusion
Hypothesis Testing : Large Samples 2
Decision Rule : If ZCAL > Z (^) α Reject H (^0) Else Accept H 0.
Hypothesis Testing For Difference Between Mean Values of Two Populations Consider two independent large random samples of size n (^) 1 and n (^) 2 drawn from two populations. Let their sample means be and. Let μ 1 and μ 2 be respective population means. The test statistic follows normal distribution for a large sample due to Central Limit Theorem. Thus: Test Statistic : If σ 1 and σ 2 are not known, the standard error of –is estimated as:
The null hypothesis that the two population means are equal is stated as: H 0 : μ 1 = μ 2 and H 1 : μ 1 ≠ μ 2. For a two-tailed test: Decision Rule: If ZCAL < – Z (^) α/2 or ZCAL > Zα/2 Reject H (^0) Else Accept H 0.
p -Value Approach for Hypothesis Testing of Single Population Mean p -value approach, also referred to as Observed Level of Significance measures the smallest level at which null hypothesis can be rejected. The p- value approach has an advantage that p -value can be directly compared to the level of significance α. Thus: Decision Rule : If p -value < α Reject H 0 Else Accept H 0.
The p -value for a two-tailed test is simply the double the critical area found in the tails of the distribution.
Relationship between Interval Estimation and Hypothesis Testing The decision rule for two-tailed test can be stated as: If Hypothesized value μ 0 lies in the confidence interval Accept H (^0) Else Reject H 0.
Hypothesis Testing For Population Proportion The decision rules for accepting or rejecting a null hypothesis for population mean both for two-tailed and one-tailed tests also apply for hypothesis testing of a population proportion.
Exercise Tick the correct option.
- Which of the following is true? A Null hypothesis is not tested but alternative hypothesis is tested, B Null hypothesis is tested but the alternative hypothesis is not tested. C Both null and alternative hypothesis are tested. D Both null and alternative hypothesis are not tested.
- Which of the following is correct? A Null hypothesis asserts that there is no significant difference between the sample statistic and the population parameter and the difference, if any, is due to sampling fluctuation. B Null hypothesis asserts that there is significant difference between the sample statistic and the population parameter and the difference is not due to sampling fluctuation. C Alternative hypothesis asserts that there is no significant difference between the sample statistic and the population parameter and the difference, if any, is due to sampling fluctuation. D Alternative hypothesis asserts that there is significant difference between the sample statistic and the population parameter and the difference is not due to sampling fluctuation.
- Level of significance is the: A minimum probability of rejecting the null hypothesis. B maximum probability of rejecting the null hypothesis. C minimum probability of rejecting the alternative hypothesis.
Hypothesis Testing : Large Samples 4
D maximum probability of rejecting the alternative hypothesis.
- Test statistic refers to a function of
A Population items B Sample observations C Either A or B D None of these
- Critical value is the value of …… which separates the critical region from the acceptance region. A Parameter B Statistic C Either A or B D None of these
- The size of critical region indicates the probability of A rejecting a true null hypothesis. B accepting a true null hypothesis. C accepting a false null hypothesis D accepting a true alternative hypothesis
- The critical value of test statistic Z at 5% level of significance for one tailed test is: A 1.96 B 2.85 C 2.33 D 1.
- Type I error is the error committed by the test in A accepting a true null hypothesis B accepting a false null hypothesis C rejecting a true null hypothesis D rejecting a false null hypothesis
- Type II error is the error committed by the test in A accepting a true null hypothesis B accepting a false null hypothesis C rejecting a true null hypothesis D rejecting a false null hypothesis
- Power of the test is the probability of: A accepting a true null hypothesis B accepting a false null hypothesis C rejecting a true null hypothesis D rejecting a false null hypothesis
- Power of a test is A Probability of Type I error B Probability of Type II error C 1 – Probability of Type I error D 1 – Probability of Type II error
- Degrees of freedom in case of a sample of size 50 is A 50 B 49 C 48 D None of these
- Degrees of freedom in case of two samples of size 50 and 60 is A 110 B 109 C 108 D None of these
- The sample size for estimating a mean is given by: A Probability of Type I error B Probability of Type II error C 1 – Probability of Type I error D 1 – Probability of Type II error
- For a particular hypothesis test α = 0.05 and β = 0.10. The power of this test is: A 0.15 B 0.95 C 0.85 D 0.
- When null hypothesis is given as – H 0 : μ < 50, the alternative hypothesis can be: A H 1 : μ ≥ 50 B H 1 : μ < 50 C H 1 : μ = 48 D H 1 : μ ≠ 50
- With a lower significance level, the probability of rejecting a null hypothesis that is actually true: A Increases B Decreases C Remains same D Cannot be said
- When a null hypothesis is accepted, it is possible that:
Business Statistics & Data Processing
- A hypothesis test is being performed for a process in which a type-I error will be very costly, but a Type-II error will be relatively inexpensive and unimportant. Which of the following wouldbe the best choice for alpha (α) in this test? A 0.10 B 0. B 0.01 D 0. [June 2014 II]
- In the hypothesis testing procedure a researcher may commit type II error in which of the following conditions? A When the true null hypothesis is rejected B When the alternative hypothesis is accepted C When the false null hypothesis is accepted D When the true null hypothesis is accepted [Dec 2014 III]
- Consider the following statements and identify the wrong statements : Statement-I : Accepting null hypothesis, when it is false, is called a level of significance. Statement-II : is called power of a test. Statement-III : Critical value of Z-static for two tailed test at 5% level of significance is 1.96. Codes : A Statement I, II and III B Statements I and III C Statements II and III D Statements I and II [June 2015 III]
- Identify which of the following step would be included in hypothesis testing : (a) State the null and alternative hypothesis (b) Set the significance level before the research study. (c) Eliminate all outliers. (d) Obtain the probability value using a computer program as such as SPSS. (e) Compare the probability value to the significance level and make the statistical decision. Codes : A (a), (b), and (d) B (c), (d) and (e) C (a), (b), (d) and (e) D (b), (c), (d) and (e) [Dec 2015 II]
- Statement-I : When a null hypothesis gets rejected in statistical hypothesis testing, it is known as II type- error in hypothesis testing. Statement-II : When a sample is small sized and parametric value of the standard deviation is not known z -test is the most appropriate test for hypothesis testing. Codes : A Both statement are correct B Both statement are incorrect C Statement-11 is correct while statement-II is incorrect D Statement-I is incorrect while statement-II is correct [Dec 2015 III]
- The power of the statistical hypothesis testing is denoted by : A α B β C 1 – α D 1 – β [Dec 2015 III]
- Read the following statements and choose the correct code : Statement – I : In statistical hypothesis-testing, null hypothesis states that there is a difference between population parameter and sample statistic. Statement – II : The two-tailed test in hypothesis testing is a non-directional test. Codes : A Both the statements are true. B Both the statements are false. C Statement – I is true while Statement – II is false. D Statement – I is false while Statement – II is true. [Sept 2016 III]
- Match the items of List-I with items of List-II and choose the correct code :
Business Statistics & Data Processing
List-I List-II (a) The ability of test to reject a null hypothesis when it is false (i) Level of significance (b) The probability of accepting a false hypothesis (ii) Type I error (c) the probability of rejecting a true null hypothesis due to sampling error (iii) Type II error (d) the probability of rejecting a true null hypothesis (iv) Power of a test Codes : a b c d
A iv ii i iii
B iv iii i ii
C i ii iii iv
D ii iii i iv
[Jan 2017 II]
Hypothesis Testing : Large Samples 8
