In Lesson Ten you’ve seen how to use a Goodness of Fit Test, a Test for Independ
ID: 3270066 • Letter: I
Question
In Lesson Ten you’ve seen how to use a Goodness of Fit Test, a Test for Independence and an Analysis of Variance to answer research questions relevant to those statistical analysis procedures. Respond to the following to demonstrate your grasp of these:
1.Describe a scenario where a researcher could use a Goodness of Fit Test to answer a research question. Fully describe the scenario and the variables involved and explain the rationale for your answer. Why is that test appropriate to use?
2.Describe a scenario where a researcher could use a Test for Independence to answer a research question. Fully describe the scenario and the variables involved and explain the rationale for your answer. Why is that test appropriate to use?
3.Describe a scenario where a researcher could use an Analysis of Variance to answer a research question. Fully describe the scenario and the variables involved and explain the rationale for your answer. Why is that test appropriate to use?
4.The Goodness of Fit Test and Test for Independence both use the same formula to calculate chi-square. Why? I.e., explain the logic of the test.
5.ANOVA is based on an F-ratio that is calculated as the ratio of two variance estimates, the variance between groups and the variance within groups, but enables conclusions to be made about the means of the samples involved. What is the logic of that? I.e., explain the rationale that supports the use of variance estimates.
Explanation / Answer
Question 1)
Chi-square test of goodness fit:
The following figures show the distribution of digits in numbers chosen at random from a telephonic dictionary.
Digits
0
1
2
3
4
5
6
7
8
9
Total
Frequency
1001
1004
900
700
950
1200
977
1027
1090
1277
10126
Test whether the digits may be taken to occur equally frequently in the directory.
The above example is a test of goodness of fit as here we are interested to check whether the observed values are going to fit to a particular pattern (equally) or not.
Question 2)
Chi-square test of independence:
Two sample polls of votes for two candidates A and B for a public office are taken one from among the residents of rural areas. The results are given in the table. Examine whether the nature of the area is related to voting preference in this election.
Area
Votes For
Total
A
B
Rural
770
290
1060
Urban
450
417
867
Total
1220
707
1927
The above example is the case of test of independence. Here we have two categorical data set, ‘Votes For’ and ‘Area’. Here we are going to check whether these two variables are independent or dependent of each other.
Question 3)
Analysis of Variance:
A manufacturing company has purchased three new machines of different makes and wishes to determine whether one of them is faster than the others in producing certain output. Five hourly production figures are observed at random from each machine and the results are given in Table. At 5% level of significance test whether the machines are significantly different in there mean speeds.
Machine A1
Machine A2
Machine A3
27
42
39
25
30
37
37
29
27
39
27
36
47
25
34
Here we are going to compare more than two population means so this is the case of analysis of variance. Here the dependent variable is the speeds and the independent variable is the types of machines.
Question 4)
The goodness of fit test and the test of independence in both the cases we use the same formula as in both cases we have the categorical values and we have observed frequency and expected frequency. Initially we have single column for test of goodness of fit and we have contingency table for test of independence but finally we can convert the data in only one column as observed frequency so we follow the same steps of calculations.
Digits
0
1
2
3
4
5
6
7
8
9
Total
Frequency
1001
1004
900
700
950
1200
977
1027
1090
1277
10126
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