Question

In some countries, people believe that blood type has a strong impact on personality. For example, Type B blood is thought to be associated with passion and creativity. A statistics student at a large U.S. university decides to test this theory. Reasoning that people involved in the arts should be passionate and creative, she takes a simple random sample of students majoring or minoring in arts at her university and asks them for their blood type. Here are her results (12 points). Observed number of performing arts majors with each blood type Type A Type B Type AB Type 0 Total 50 23 10 67 150 Assume the distribution of blood type among all U.S. residents is as follows: Type A: 42%; Type B: 10%; Type AB: 4%; Type O: 44%. (a) The student wants to carry out a test of significance to see if the distribution of blood types among arts majors or minors is different from the U.S. distribution. State the null and alternative hypotheses for this test. (2 points) (b) Find the expected counts for each blood type under the assumption that the null hypothesis is true. (2 points) Expected number of performing arts majors with each blood type Type A Type B Type AB Type 0 Total (c) Discuss whether the conditions for this test have been met. (3 points) (d) Find the value of the test statistic and the P-value of the test, and make the appropriate conclusion. Use α = 0.05. (5 points) (e) Based on your answer to (d), which error is it possible that you have made, Type I or Type II? Describe that error in the context of the problem. (2 points extra credit).

Answer 1

a.

Hypothesis:

H0: The distribution of blood types among arts majors or minors is same as the U.S. distribution.

Ha: The distribution of blood types among arts majors or minors is different from the U.S. distribution.

b.

Observation table

Blood type	Oi	Pi	Ei=Npi	(Oi-Ei)2/Ei
A	50	0.42	63	2.68253968
Blood type	23	0.1	15	4.26666667
AB	10	0.04	6	2.66666667
O	67	0.44	66	0.01515152
	150		150	9.63102453

c.

All expected frequencies are greater than 5, hence condition is satisfy.

d.

Test statistic,

= 9.6310

Degrees offreedom = n-s-k-1 = 4-0-0-1 = 3

Where,

n = Total number of observations

s = number of parameter estimated

k= number of degrees of freedom lost due to pooling

P-value = 0.022

It is obtained using Excel with command =CHIDIST(I15,3)

Since p-value is less than 0.05, we reject null hypothesis and conclude that The distribution of blood types among arts majors or minors is different from the U.S. distribution.

e.

Type I error : Rejecting nullhypothesis, when it is true.

Type II error : Accepting null hypothesis wehn it is false.

Here we reject null hypothesis, hence it may possible that we made type I error.

In this context type I error means actually The distribution of blood types among arts majors or minors is same as the U.S. distribution, but we conclude that it is different.