Question

An astronomer is investigating whether or not there is a relationship between the type of a planet and the shape of its orbit. There are three types of planets: rocky, gas giant, and ice giant, and two types of orbit: circular and elliptical. The astronomer gets information on observed planets and lists their characteristics, they assume that the planets that have been observed are a simple random sample of all planets. Counts for the different kinds of planets are given below

	Rocky	Gas Giant	Ice Giant	Total
Circular Orbit	48	62	53	163
Elliptical Orbit	99	102	79	280
Total	147	164	132	443

a) State the astronomer's hypotheses.

b) Compute the test statistic and p-value for this test.

c) Reach your conclusion for the test at the .05 significance level.

d) The astronomer has a theory that fewer than half of all rocky planets have circular orbits. What kind of statistical test could be used to answer that question.

Answer 1

a)

The null and alternate hypothesis are:

H0: There is no relationship between the type of a planet and the shape of its orbit.
Ha: There is a relationship between the type of a planet and the shape of its orbit.

b)

Observed (Oi):

	Rocky	Gas Giant	Ice Giant	TOTAL
Circular orbit	48	62	53	163
Elliptical orbit	99	102	79	280
TOTAL	147	164	132	443

Now, Expected frequency = [(Row total) x (Column total)] / Table total

Expected (Ei):

	Rocky	Gas Giant	Ice Giant	TOTAL
Circular orbit		60.343	48.569	163
Elliptical orbit	92.912	103.657	83.431	280
TOTAL	147	164	132	443

Test statistic value =

The p-value is given by:

c)

Since p-value is greater than 0.05, so we do not have sufficient evidence to reject the null hypothesis H0.

Thus we conclude that there is no relationship between the type of a planet and the shape of its orbit.

d)

This a test for true proportion. So, a z-test should be used.