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MANGALORE INSTITUTE OF TECHNOLOGY & ENGINEERING

(A Unit of Rajalaxmi Education Trust ®, Mangalore)

Autonomous Institute affiliated to VTU, Belgavi, Approved by AICTE, New Delhi

Accredited by NAAC with A+ Grade and ISO 9001:2015 Certified Institution

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CONTENT

Sampling Theory: Introduction, Population, Sampling, Parameter and Statistic, Errors in Statistics, Simple Random Sampling.
Testing of Hypothesis: Introduction, Sampling Distribution of a Statistic, Central Limit theorem, Test of Significance, Null Hypothesis, Alternative Hypothesis, Types of Errors in Testing of Hypothesis, Level of Significance, Critical Region, One-tailed and Two-tailed tests, Procedure for Testing of Hypothesis.
Parametric tests: Test of Significance of single Mean and difference of Mean. Student’s ‘t’ Distribution, Non-parametric tests: Chi-Square test, Wilcoxon Signed-Rank test.

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MODULE -2:

STATISTICAL INFERENCE AND HYPOTHESIS TESTING

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RBT LEVEL: L1, L2, L3

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LEARNING OBJECTIVES:

Impart knowledge of Statistical Inference and Hypothetical Testing.

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Population:

In a statistical investigation, the study of variation with respect one or more characteristics relating to individuals belonging to a group. This group of individuals under study is called population or Universe.
The population may be finite or infinite.
Example: If we want to have an average per capita(monthly) income of the people in India, we will have to enumerate all the earning individuals in the country, which is rather a very difficult task.

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Sample:

A finite subset of statistical individuals in a population is called a sample.
A housewife normally tests the cooked products to find if they are properly cooked and contain the proper quantity of salt.

For Example, in a shop we assess the quality of sugar, wheat or any other commodity by taking a handful of it from the bag and then decide to purchase it or not.

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Parameter and Statistic:

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Sl. No.	Population	Sample
1	Statistical Constant of the population are referred to as ‘parameters’.	Statistical Constant of the sample are referred to as ‘statistics’.
2	Mean: 𝝁
3	Variance: 𝝈²	Variance: s²
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Random Sampling:

A Random sample is one in which each unit of population has an equal chance of being included in it.

Suppose we take a sample size n from a finite population of size N. Then there are ^NC_n possible samples.

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Sample Number
1
2
3
. . .	. . .	. . .	. . .
k

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Null and Alternative Hypothesis:

A hypothesis (assumption) -a definite statement about the population parameter. Such a hypothesis, which is usually a hypothesis of no difference, is called null hypothesis and is usually denoted by H_0.

For example, the average test score of students in a particular class is equal to the school-wide average. If we want to test the null hypothesis that the population (school-wide average 𝝁 ) has a specified mean 𝝁₀ (average test score of students in a particular class ) say, H₀: 𝝁 =𝝁₀

Then the alternative hypothesis could be :

i) H₁: 𝝁 ≠𝝁₀(i. e., : 𝝁 >𝝁₀or: 𝝁 < 𝝁₀)

ii) H₁: 𝝁 >𝝁₀

iii) H₁: 𝝁 < 𝝁₀

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According to Prof. R. A. Fisher, null hypothesis is the hypothesis which is tested for possible rejection under the assumption that it is true.

For example, consider a person in court who is charged with murder. The jury needs to decide whether the person is innocent (the null hypothesis) or guilty (the alternative hypothesis). As usual, we assume the person is innocent unless the jury can provide sufficient evidence that the person is guilty. Similarly, we assume that H₀ is true unless we can provide sufficient evidence that it is false and that H₁ is true, in which case we reject H₀ and accept H₁.

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H₀: A person is innocent

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One-tailed and Two-tailed Tests:

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Test of Significance:

The process which help us to decide about the acceptance or rejection of the hypothesis is called the Test of significance
Since, for large n, almost all the distributions, e.g., t, F, Chi-square, can be approximated very closely by a normal probability curve, we use the Normal Test of Significance for large samples.
Some of the well-known tests of significance for small samples are t-test, F-test.

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In a particular problem, whether one-tailed or two-tailed test is to be applied depends entirely on the nature of the alternative hypothesis. If the alternative hypothesis is two-tailed, we apply two-tailed test and if alternative hypothesis is one-tailed, we apply one-tailed test.

Example:

Suppose that there are two population brands of bulbs, one manufactured by standard process (with mean life 𝝁₁) and the other manufactured by some new technique (with mean life 𝝁₂). If we want to test if the bulb differ significantly, then our null hypothesis is H₀: 𝝁₁ = 𝝁₂ and alternative will be H₁: 𝝁₁≠𝝁₂, thus giving us two-tailed test.
However, if we want to test if the bulbs produced by new process have higher average life than those produced by standard process, then we have H₀: 𝝁₁ = 𝝁₂ and H₁: 𝝁₁< 𝝁₂, thus giving us a left-tailed test.
Similarly, for testing if the product of new process is inferior to that of the standard process, then we have H₀: 𝝁₁ = 𝝁₂ and H₁: 𝝁₁> 𝝁₂, thus giving us a right-tailed test.

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	1%	5%
Two-tailed test
Right-tailed test
Left-tailed test

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Test of significance for single mean

Test of significance for difference of means

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Age last birthday
16-20	12	18	100	1200
21-25	22	23	25	550
26-36	20	28	0	0
31-35	30	33	25	750
36-40	16	38	100	1600
Total

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Graph of t-distribution

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Sample Standard Deviation is given
Sample Standard Deviation is not given	For individual data
	For comparative datas
Mean and Standard Deviation is given directly

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70	-27.2	739.44
120	22.8	519.84
110	12.8	163.84
101	3.8	14.44
88	-9.2	84.64
83	-14.2	201.64
95	-2.2	4.84
98	0.8	0.64
107	9.8	96.04
100	2.8	7.84
Total 972	0	1833.60

Table 1: Calculation for Sample Mean and S.D. :

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x			y
25	-3	9	44	14	196
32	4	16	34	4	16
30	2	4	22	-8	64
34	6	36	10	-20	400
24	-4	16	47	17	289
14	-14	196	31	1	1
32	4	16	40	10	100
24	-4	16	30	0	0
30	2	4	32	2	4
31	3	9	35	5	25
35	7	49	18	-12	144
25	-3	9	21	-9	81
			35	5	25
			29	-1	1
			22	-8	64

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Sailors			Soldiers

63	-5	25	61	-5	25
65	-3	9	62	-4	16
68	0	0	65	-1	1
69	1	1	66	0	0
71	3	9	69	3	9
72	4	16	69	3	9
			70	4	16
			71	5	25
			72	6	36
			73	7	49
Total	0	60	Total	18	186

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	Mean	Standard deviation
Alloyed wire	0.083 ohm	0.003 ohm
Standard wire	0.136 ohm	0.002 ohm

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		1	2	3	4	5	6	7	8	Total
Increase weight in lb	Food A	49	53	51	52	47	50	52	53	407
Increase weight in lb	Food B	52	55	52	53	50	54	54	53	423

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	Type I	Type II
Sample No.
Sample Means
Sample S.D. ’s

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Exact Sampling Distribution:

Paramteric tests
Non-parametric methods or distribution-free methods

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Days :	Mon	Tue	Wed	Thu	Fri	Sat
No. of parts demanded :	1124	1125	1110	1120	1126	1115

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Frequency
Days
Mon	1124	1120	16	0.014
Tue	1125	1120	25	0.022
Wed	1110	1120	100	0.089
Thur	1120	1120	0	0
Fri	1126	1120	36	0.032
Sat	1115	1120	25	0.022
Total	6720	6720		0.179

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Frequency
Category
Failed	46		1156	14.450
III Division	68		64	1.067
II Division	62		484	12.100
I Division	24		16	0.800
Total	200	200		28.417

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No. of boys	0	1	2	3	4
No. of girls	4	3	2	1	0
No. of families	32	178	290	236	64

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No. of male births	Frequency (Families)
No. of male births
0	32	50	324	6.48
1	178	200	484	2.42
2	290	300	100	0.33
3	236	200	1296	6.48
4	64	50	196	3.92
Total	800	800		19.63

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No. of mistakes in a page (x)	0	1	2	3	4	5	6
	275	72	30	7	5	2	1

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Face	1	2	3	4	5	6
Frequency	8	7	12	8	14	11

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Digits	0	1	2	3	4	5	6	7	8	9	Total
Frequency	1026	1107	997	966	1075	933	1107	972	964	853	10,000

Area	Votes for		Total
Area	A	B	Total
Rural	620	380	1000
Urban	550	450	1000
Total	1170	830	2000

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Days :	Mon	Tue	Wed	Thu	Fri	Sat	Total
No. of accidents:	14	18	12	11	15	14	84

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x			y
25	-3	9	44	14	196
32	4	16	34	4	16
30	2	4	22	-8	64
34	6	36	10	-20	400
24	-4	16	47	17	289
14	-14	196	31	1	1
32	4	16	40	10	100
24	-4	16	30	0	0
30	2	4	32	2	4
31	3	9	35	5	25
35	7	49	18	-12	144
25	-3	9	21	-9	81
			35	5	25
			29	-1	1
			22	-8	64

x			y
25	-3	9	44	14	196
32	4	16	34	4	16
30	2	4	22	-8	64
34	6	36	10	-20	400
24	-4	16	47	17	289
14	-14	196	31	1	1
32	4	16	40	10	100
24	-4	16	30	0	0
30	2	4	32	2	4
31	3	9	35	5	25
35	7	49	18	-12	144
25	-3	9	21	-9	81
			35	5	25
			29	-1	1
			22	-8	64

x			y
25	-3	9	44	14	196
32	4	16	34	4	16
30	2	4	22	-8	64
34	6	36	10	-20	400
24	-4	16	47	17	289
14	-14	196	31	1	1
32	4	16	40	10	100
24	-4	16	30	0	0
30	2	4	32	2	4
31	3	9	35	5	25
35	7	49	18	-12	144
25	-3	9	21	-9	81
			35	5	25
			29	-1	1
			22	-8	64