Given a sample of 3200 data points, you need to test if the data come from a nor

Question

Given a sample of 3200 data points, you need to test if the data come from a normal population at alpha = 0.04. Based on the sample mean and sample standard deviation, the original data are transformed into z-scores with the observed frequencies as in the table below. Give an equation that shows how the original data are transformed into z-scores. Form the null and alternative hypotheses H_0 and H_1. Make a sketch of the test. Specify appropriate equations for the X^2 statistic and degrees of freedom df. Calculate the remaining probabilities with Excel lookups in Hint. Based on the expected probabilities, calculate expected frequencies. Based on expected and observed frequencies, calculate the X^2 statistic and its df. Given alpha = 0.04 and df, use CHlSQ. lNV(*, *) to look up for the critical value X^2*. Compare X^2 statistic with X^2* critical to reject or accept H_0. Interpret your result. Sketch and find the p-value of the test. Would you reject the null if alpha = 0.01? Implement parts (d) through (i) in Excel. Give a printout of the Excel worksheet.

Explanation / Answer

(a) using z=(x-mean)/sd has been used to change the original data into z-score data

(b) null hypothesis H0: observed frequency=expected frequency

and alternate hypothesis observed frequency is not equal to expected frequency

(c) the chi-square=sum(O-E)2/E) with k-1=7-1=6 df

(d) (e) and (f)

(f) chi-square=16.704

(g)critical chi-squar(0.04,6)=13.2

(h) critical chi-squar=13.2 is less than calculated chi-square=16.704 so fail to accept H0 and conclude that oberved frequency is not equal to expected frequency

(i)p-value=0.0104 ( using ms-excel command =CHIDIST(16.704,6))

since p-value is less than alpha=0.01 so fail to reject H0

category z-interval expected probability expected frequency(E) observed frequency(O) (O-E) (O-E)2/E 1 (-,-2.5) 0.006 19.871 25.000 5.129 1.324 2 (-2.5,-1.5) 0.061 193.912 201.000 7.088 0.259 3 (-1.5,-0.5) 0.242 773.537 849.000 75.463 7.362 4 (-0.5 ,0.5) 0.383 1225.360 1197.000 -28.360 0.656 5 (0.5,1.5) 0.242 773.537 751.000 -22.537 0.657 6 (1.5,2.5) 0.061 193.912 162.000 -31.912 5.252 7 (2.5,) 0.006 19.871 15.000 -4.871 1.194 1.000 3200.000 3200.000 0.000 16.704

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