Question

The numbers 1,6,15,20,15,6,1 are the coefficients of the binominal expansion (p+q)^6. Use the normal quantile plot method to show that these numbers are close to a normal distribution.

Answer 1

first sort the values in ascending order

1

1

6

6

15

15

20

here we have 7 values so divide the standard normal distribution into 8 equal-sized areas with 7 values. so each segment has 12.5% area

find z values for each of the segment

	probability	z values
12.5	0.125	-1.15035
25	0.25	-0.67449
37.5	0.375	-0.31864
50	0.5	-1.4E-16
62.5	0.625	0.318639
75	0.75	0.67449
87.5	0.875	1.150349

i have used excel to find out z scores(NORMSINV)

now plot the graph with the following values

actual quantiles	theoretical quantiles
1	-1.15034938
1	-0.67448975
6	-0.318639364
6	-1.39214E-16
15	0.318639364
15	0.67448975
20	1.15034938

here R-value is 0.916. straight line is good fit for data

straight line on q q plot indicates the data is approximately normal.

The numbers 1,6,15,20,15,6,1 are close to a normal distribution.

remark:i have used excel to draw the graph