the 50 values listed below are intended to be a random sample from the standard

ID: 3180996 • Letter: T

Question

the 50 values listed below are intended to be a random sample from the standard normal distribution

(1) carry out a X2 test of goodness of fit by dividing the real line into 5 intervals, each of which has probability 0.2 under the standard normal distribution.

(2) carry out a X2 test of goodness of fit by dividing the real line into 10 intervals, each of which has probability 0.1 under the standard normal distribution.

1.28 -1.22 35 32 8 1.66 1.39 38 1.38 -1.26 .49 85 2.33 34 -1.96 64 -1.32 -1.14 .64 3.44 -1.67 .85 41 1.13 67 41 04 1.24 1.05 .04 .76 61 -2.04 35 2.82 46 63 -1.61 13 -1.81 11

Explanation / Answer

Goodness of Fit [Since level of significance is not stipulated in the question, the most frequently used level of 5% is used for this calculation]

Let Oi and Ei be respectively the observed frequency and expected frequency of the ith class. Then, the test statistics for Null Hypothesis, H0: the given data fits into Standard Normal Distribution Vs HA: H0 is false, is

Chi-square = sum over i of {(Oi - Ei)2/Ei} and under H0, it is distributed as Chi-square with (k - 1) degrees of freedom, where k = number of classes.

Decision criterion: reject H0, if Chi-squarecal > Chi-squarecrit

Calculations are shown in the table given below.

Chi-squarecal = 7.4

Chi-squarecrit = upper 5% point of Chi-square with DF = 4 = 9.49

Since Chi-squarecal < Chi-squarecrit , H0 is accepted.

Class

Probability

(Oi - Ei )^2/Ei

Less than - 0.842

0.2

3.6

[- 0.842 - - 0.253)

0.2

[- 0.253 - 0.253)

0.2

0.9

[0.253 - 0.842)

0.2

0.4

0.842 and greater

0.2

2.5

Total

7.4