CHI- SQUARE TEST
In sampling testing certain assumptions are made about the population and samples. For example in sampling distribution it is assumed that the samples are drawn from a population following normal distribution. But sometimes it is not possible to make any assumptions about the distribution of the population from where the samples are taken. In such situations we follow a non – parametric test. Chi Square is one such non – parametric test
While collecting data if the observed frequency ‘O’ and expected frequency ‘E’ are known then
ᵡ
2 = ∑ *(O – E) 2 /E] follows a distribution known as chi square distribution. Properties of Chi Square Test1. It is a non – parametric test. Assumptions about the form of the distribution or its parameters are not required
2. It is a distribution free test, which can be used in any type of distribution of population. 3. It is easy to calculate chi – square statistics
4. It analyses the differences between a set of observed frequencies and a set of corresponding expected frequencies.
5. It is a multinomial distribution 6. The variable varies from 0 to ∞ 7. It is a one tailed test
Uses of Chi Square Test
1. Useful for the test of goodness of fit
2. Useful for the test of independence of attributes 3. Useful for testing homogeneity
4. Useful for testing given population variance Conditions for Applying Chi Square Test
1. The total frequencies (N) must be reasonably large than or at least 50
2. Observed frequency of less than 5 is pooled with the preceding or succeeding frequencies so that no observed frequency is less than 5. Then the degree of freedom is based on the resulting number of frequencies.
3. The distribution should not be expressed as proportions or percentages. It should be of original units.
How to Test the Goodness of Fit for a Multinomial Distribution? 1. State the null and alternate hypothesis
2. Select a random sample and record the observed frequencies (Oi) for each category.
3. Assume the null hypothesis is true. With this assumption determine the expected frequencies (Ei).
4. Compute the value of the test statistic using
ᵡ
2 = ∑ *(Oi – Ei) 2 /Ei]5. Fix the level of significance 6. Write the degree of freedom
7. Find the critical value corresponding to the degree of freedom and level of significance. 8. Check whether
a. Test statistic is < Critical Value b. Test statistic is > Critical Value
Accept the null hypothesis if
ᵡ
2 (Test Statistic) < Critical ValueReject the null hypothesis if
ᵡ
2 (Test Statistic) > Critical Valueᵡ
2Distribution
Type - I
1. Explain
ᵡ
2
test. How will you use this to test a hypothesis? What are the precautions which are to be taken for
ᵡ
2testing? Mention the uses ofᵡ
2.
2. The following data shows the distribution of frequency between educational level and awareness of AIDS. Find the relationship between education and awareness.
Awareness Level
Education Low Moderate Higher
Illiterate 320 50 30
Primary 80 15 05
Middle School 110 70 20
High School 200 60 40
3. Due to recession, an IT company is planning to lay-off some of its personnel. The results of opinion survey conducted by a company are given below. Formulate the hypothesis and test it using
ᵡ2
test at 0.05 level of significanceOpinion Towards Retrenchment
Occupation Oppose Undecided Favor
Administrative Staff 37 16 19
Project Team Manager 46 22 15
Project Team Leader 32 11 2
4. The following table gives the data regarding the field of study in the University and their field of Specialization in High School
Field of Study in The Unversity Specialization in High
School
Biology Medicine Agriculture
Biology 26 52 23
Physics & Mathematics 3 44 8
Agriculture 4 1 1
Humanities 6 4 10
Test whether there is any association between High School Specialization and field of study in the University
5. Based on the following data test the hypothesis that there is no difference in quality of the kind of tyres. (α = 0.05)
Tyre Brand
A B C D
Failed to last 4000 kms 26 23 15 32
Lasted to 4000-6000kms 118 93 116 121
Lasted for more than 6000kms 56 84 67 49
6. Following table provides the number of executives according to the time devoted to public activities by rank
Rank
Time Devoted Manager Sr. Manager Gr.Manager
A Good Deal 25 13 9
Some Time 62 53 49
Test the hypothesis that the time devoted to public activities is independent of the rank. 7. A survey was conducted in Bangalore city as well as in the rest of Karnataka state regarding the people’s first choice of four types of magazines. The results are tabulated below
City
Type of Magazine Bangalore Rest of Karnataka
News Magazine 70 310
Movie Magazine 60 280
Ladies Magazine 40 170
Sports Magazine 30 40
Test whether there is any significant difference between Bangalore population and rest of Karnataka in the choice of magazines.
8. A sample of 115 professionals, 110 businessmen, and 125 farmers were chosen and asked to express their feelings regarding a national policy. The result of the survey is given below:
Occupation Favorable Against Indifferent
Professionals 80 21 14
Businessmen 72 15 23
Farmers 69 31 25
Can we conclude that there exists no difference in opinion among the three classes of people in the policy?
9. A survey on the consumption of alcoholic drinks by menial staff of a municipal corporation gave the following data:
Consumers Non - Consumers
Attenders 86 31
Scavengers 46 26
Road Workers 64 10
Test whether there is any association between type of work and alcoholic drinking habit
10. The results of a survey to know the educational attainment among 100 persons randomly selected in a locality are given below:
Education
Sex Middle School High School College
Male 10 15 25
Female 25 10 15
11. The following table gives a sample of married women, their level of education and marriage adjustment scores.
Marriage Adjustment Scores Level of
Education
Low Medium High
College 24 97 62
High School 26 32 24
Middle School 40 22 14
Analyze the data and give your comments
12. Formulate an appropriate hypothesis and use
ᵡ
2
test for the following data Inter-Caste Marriage
Socioeconomic Status Favorable Indifferent Unfavorable
Low 40 25 10
Moderate 35 30 15
High 25 45 5
13. An oil company has explored three different areas for possible oil reserves the results of the test were given as below.
Area
A B C
Strikes 7 10 8
Dry Holes 10 18 9
Does the data suggest that the three areas have the same potential at 10% level of significance?
Type - II
14. On the basis of the information given below find if there is any association with inoculation and absence of attack of typhoid at 5% level of significance.
Attacked Not Attacked Total
Inoculated 12 674 686
Not Inoculated 47 1122 1169
15. Find the relationship between Educational Status and Safety awareness of workers, Use
ᵡ
2 test at 0.05 level of significance. Propose the Null hypothesis and Alternate Hypothesis.Level of Awareness
Educational Status Low High
Upto 10th Std. 30 70
Professional 60 40
Type – III
16. A theory predicts the proportion of beans in the four groups A, B, C and D should be 9:3:3:1. In an experiment among 1600 beans, the numbers in the four groups were 882, 313, 287 and 118. Does the experiment result support the theory? Apply
ᵡ
2
test.
17. A college is running post graduate classes in five subjects with equal number of students. The total number of absentees in these five classes is 75. Test the hypothesis that these classes are alike in absentees if the actual absentees in each are as follows: History = 9; Philosophy = 18; Economics = 15; Commerce = 12; Chemistry = 11.
18. How well do the airline companies serve their customers? A study showed the following customer ratings. 3% Excellent, 28% Good, 45% Fair and 24% Poor. In a follow up study of telephone companies, a sample of 400 adults found the following customer ratings 24 Excellent, 124 Good, 172 Fair and 80 Poor. Does the distribution of customer ratings for telephone companies differ from the distribution of the customer ratings for the airline companies?
19. A multinomial population with four categories A, B, C and D have the following proportion of items same in all categories. A sample of size 300 yielded the following results: A = 85, B = 95, C = 50 and D = 70. Use α = 0.05 to determine whether the claim of the proportions being same in every category is true.
20. During the first 13 weeks of television season, the Saturday evening 8.00pm to 9.00pm audience proportion were recorded as ABC = 29%, CBS = 28%, NBC = 25% and Independent = 18%. A sample of 300 homes two weeks after a Saturday night schedule revision yielded the following viewing audience data ABC = 95 homes, CBS = 70 homes, NBC = 89 homes and Independent = 46 homes. Test with α = 0.05 to determine whether the viewing audience proportion have changed.
Type – IV
21. A die is thrown 132 times with the following results
No: Turned Up 1 2 3 4 5 6
Frequency 16 20 25 14 29 28
Is the die biased?
22. Eight coins were tossed 256 times and the following results were obtained
No: of Heads 0 1 2 3 4 5 6 7 8
Frequency 2 6 30 52 67 56 32 10 1
Is the coin biased? Use
ᵡ
2
test
23. The following shows the result of throwing 12 dice, 4096 times. A throw of 4,5,6 being called a success. Fit a Binomial Distribution.
Success 0 1 2 3 4 5 6 7 8 9 10 11 12
Frequency - 7 60 198 430 731 948 847 536 257 71 11 - Test for goodness of fit.
24. A typist kept a record of mistakes made per day during 300 working days in a year.
Mistakes/Day 0 1 2 3 4 5 6
No: of Days 143 90 42 12 9 3 1
Fit a Poisson distribution to the data. Test for goodness of fit. (Given
=
0.410652)
25. The following are the number of arrivals of flights per hour in an airport. Can we conclude that the following 400 arrivals follow a Poisson distribution with at 5% level of significance?
No: of Arrivals 0 1 2 3 4 5
