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# Basic Large Sample Theory

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Question:
For i = 1,...,s and j = 1,...,ni , let Xij be independent with Xi,j having distribution Fi , where Fi is
an arbitrary distribution with mean i and finite variances 2 that vary for each Fi......

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#### Solution Preview

Robustness of F- statistic under unequal variances
(A separate Excel working is attached for the development of Rejection probabilities and it graph)
The F-statistic that we consider in this exercise is nothing but the ratio of "Mean Between Sum of Squares" and "Mean Within Sum of Squares". If this ratio is sufficiently large, we reject the null hypothesis. The distribution of this ratio is derived based on the fact that its denominator and numerator follow independent Chi-square distributions. Again the assumption of chi-square distribution requires the assumption that the samples are taken from Normal distribution with almost similar variances.
In this exercise we are going to verify how this F-statistic is robust when the assumption of equal variances is violated. To establish this fact we make use of a hypothetical population with the following structure.
Groups Sample size Mean Standard Deviation
1 10 55 10
2 14 80 10
3 16 62 10

The above data consists of three groups with considerably different means (Group means are not equal) and with equal standard deviation. Thus one would expect to reject the ...

#### Solution Summary

Basic large sample theory is examined for distributions. Finite variances are analyzed.

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## Basic Statistics using Excel and Eviews

See attached files for assignments.

TASK1
You are required to prepare a dataset consisting of the Income (X) in £ per week and Years of Education (Y) of 40 randomly chosen adults and then perform the following exercises. (You may enter numbers representing these characteristics of fictitious individuals, provided these are realistic).

Part A
All computer work in this section is to be performed using Microsoft Excel software. Define any term you need to use in your answer, draw appropriate diagrams and explain all relevant theory.

i. Obtain the descriptive statistics and the histogram of the data on Income and discuss whether or not standard deviation is a better measure than Semi Inter Quartile Range for this data.
ii. Now sort the data in ascending order and then delete the 5 largest and 5 smallest values. You now have 30 numbers representing income of individuals. Repeat the exercise in part (i) above for this reduced data.

Part B
All computer work in this section is to be performed using EViews software. Define any term you need to use in your answer, draw appropriate diagrams and explain all relevant theory.

iii. Obtain the descriptive statistics and the histogram of the data on Years of Education and discuss whether or not standard deviation is a better measure than Semi Inter Quartile Range for this data.
iv. Now sort the data in ascending order and then delete the 5 largest and 5 smallest values. You now have 30 numbers representing Years of Education of individuals. Repeat the exercise in part (i) above for this reduced data.

Use Excel to randomly generate 100 observations for the random variable
(See attached file).

(ii) Do your answers above suggest that the distribution of the generated numbers is approximately normal? Fully explain your answer.

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