Answer all Exercise 3. Many questionnaires use a five point scale. We practice C

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Exercise 3. Many questionnaires use a five point scale. We practice Chapter 1 ideals on a hypothetical list created from such a question. The scores to a hypothetical question of this kind, collated from 84 tests, are in the following table.

score 1 2 3 4 5 freq. 18 27 8 17 14 For this question, let c_1, ..., c_84 represent the indexed list of scores in increasing order. (A) What is c_68? (B) What is the median of the list? (C) Find (D) Find (E) What is the mean of the list? (Give your answer to the nearest thousandth). (F) What is the variance of the list? (Give your answer to the nearest thousandth).

Explanation / Answer

Part A

observations score

C1-C18 1

C19-C45 2

C46-C53 3

C54-C70 4

C71-C84 5

C68 lies in C54-C70, hence C68 is 4.

Part B

Median is the midpoint of a frequency distribution of observed values her 84/2=42

Since this list has even number of observations hence its has 2 values i.e C42 and C43 as median, which from above table lies in C19-C45 bracket ,hence median is 2.

Part C

=(C1+C2+.......+C18)+(C19+........+C45)+(C46+.....+C53)+(C54+......+C70)+(C71+..........+C84)

=18*1+27*2+8*3+17*4+14*5

=234

PART D

=(C1^2+C2^2+..............+C84^2)

=18*1^2+27*2^2+8*3^2+17*4^2+14*5^2

=820

PART E

MEAN= SUM OF OBSERVATIONS / NUMBER OF OBSERVATIONS

MEAN= 234/84=2.78

PART F

VARIANCE=(SUM ((C1-MEAN)^2+.....................(C84-MEAN)^2))/NUMBER OF OBSEVATIONS

=(18*(1-2.78)^2 + 27*(2-2.78)^2 + 8*(3-2.78)^2 + 17*(4-2.78)^2 + 14*(5-2.78)^2) / 184

=0.91

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