Question

It has been claimed that, for a penny minted in 1999 or earlier, the probability of observing heads upon spin- ning the penny is p = 0.30. Three students got together, and they would each spin a penny and record the num- ber X of heads out of the three spins. They repeated this experiment n = 200 times, observing 0, 1, 2, and 3 heads 57, 95, 38, and 10 times, respectively.

You have n = 200 data points: 0 appears 57 times, 1 appears 95 tmes, 2 appears 38 times and 3 appears 10 times. Test the hypothesis that this data came from binomial distribution. (Here you need to estimate the parameters of the binomial first.)

Answer 1

Null hypothesis :H_o : The data came from binomial distribution

Alternate hypothesis :H₁ : The data did not come from binomial distribution

Given,

X : Number of Heads when a penny is tossed

Observed Frequency:

X	O: Observed Frequency
0	57
1	95
2	38
3	10

As the penny tossed for 3 times ; n= 3

Probability of observing heads upon spinning the penny is p = 0.30

q = 1-p = 1-0.3=0.7

X : Number of Heads when a penny is tossed;

X binomial distribution with p = 0.3 and n= 3 with probability mass function

Expected frequency = Number of times the experiment conducted x P(X=x)

E = 200P(X=x)

x	P(x)	P(x)	E= 200*P(x)
0		0.3430	68.6
1		0.4410	88.2
2		0.1890	37.8
3		0.0270	5.4

Test Statistic :

O	E	O-E	(O-E)2	(O-E)2/E
57	68.6	-11.6	134.56	1.9615
95	88.2	6.8	46.24	0.5243
38	37.8	0.2	0.04	0.0011
10	5.4	4.6	21.16	3.9185
			Total	6.4054

Degrees of freedom = 4-1 =3

Level of significance = 0.05

As p-Value : 0.0934 > Level of significance: 0.05 . Fail to reject null hypothesis.

Hence Conclude that this data has come from Binomial distribution.

p-value is computed using excel function

CHISQ.DIST.RT(6.4054,3) = 0.0934

CHISQ.DIST.RT function

Returns the right-tailed probability of the chi-squared distribution.

The χ2 distribution is associated with a χ2 test. Use the χ2 test to compare observed and expected values. For example, a genetic experiment might hypothesize that the next generation of plants will exhibit a certain set of colors. By comparing the observed results with the expected ones, you can decide whether your original hypothesis is valid.

Syntax

CHISQ.DIST.RT(x,deg_freedom)

The CHISQ.DIST.RT function syntax has the following arguments:

• X     Required. The value at which you want to evaluate the distribution.

• Deg_freedom     Required. The number of degrees of freedom.