Question

one question 2 pictures

2.4.8, pp 97 101. A student counts the number of chocolate chips in cookies and records the data shown in the table below (data collected by Emily Brandt, 2010). Determine relative frequency and also , and f(x), if the number of chocolate chips is described by a Poisson distribution. I 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 frequency 1 1 1 2 2 0 0 5 5 1 0 1 1 0 0 x 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 freg 1 1 1 2 2 0 0 5 5 1 0 1 1 0 0 rel freq 0.05 0.05 0.05 0.10 0.10 0.00 0.00 0.25 0.25 0.05 0.00 0.05 0.05 0.00 0.00 f(x) 0.04 0.05 0.06 0.07 0.08 0.08 0.09 0.08 0.08 0.07 0.06 0.05 0.04 0.03 0.02 1-24.15 X 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 freq 1 1 1 2 2 0 0 5 5 1 0 1 1 0 0 rel freq 0.04 0.05 0.06 0.07 0.08 0.08 0.09 0.08 0.08 0.07 0.06 0.05 0.04 0.03 0.02 f(x) 0.05 0.05 0.05 0.10 0.10 0.00 0.00 0.25 0.25 0.05 0.00 0.05 0.05 0.00 0.00 22.2 x 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 freq 1 1 1 2 2 0 0 5 5 1 0 1 1 0 0 rel freq 0.05 0.05 0.05 0.10 0.10 0.00 0.00 0.25 0.25 0.05 0.00 0.05 0.05 0.00 0.00 f(x) 0.04 0.05 0.06 0.07 0.08 0.08 0.09 0.08 0.08 0.07 0.06 0.05 0.04 0.03 0.02 x=22.2

Answer 1

Fitting Poisson Distribution

We get the relative frequencies by dividing each frequency by the total frequency. If the number of choco chips is described by a Poisson distribution, then $\lambda$ should be equal to the mean of the sample. We get the following table:

x	frequency(f)	relative freq	xf
16	1	0.05	16
17	1	0.05	17
18	1	0.05	18
19	2	0.1	38
20	2	0.1	40
21	0	0	0
22	0	0	0
23	5	0.25	115
24	5	0.25	120
25	1	0.05	25
26	0	0	0
27	1	0.05	27
28	1	0.05	28
29	0	0	0
30	0	0	0
Total	20	1	444

So the mean is given by: $\sum xf/\sum f$ = 444/20 = 22.2.

Thus $\lambda\approx$ 22.2.

Now we will calculate the mass values.

We know that the pmf of Poisson distribution is given by: f(x) = exp(-$\lambda$).$\lambda$^$x$/ $x$!

So putting $\lambda\approx$ 22.2 we get:

x	f(x)
16	0.04
17	0.05
18	0.06
19	0.07
20	0.08
21	0.08
22	0.09
23	0.08
24	0.08
25	0.07
26	0.06
27	0.05
28	0.04
29	0.03
30	0.02

Thus third option is the correct one.