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 conclusions for the test at the 0.05 significance level. Be sure to interpret in the context of the problem.

(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

Chi-Square Independence test - Results

PART (A) (1) Null and Alternative Hypotheses The following null and alternative hypotheses need to be tested: H0​: The two variables - Type of planet and Shape of orbit are independent Ha​: The two variables - Type of planet and Shape of orbit are dependent This corresponds to a Chi-Square test of independence. PART (B) (2) Degrees of Freedom The number of degrees of freedom is df = (2 - 1) * (3 - 1) = 2 (3) Critical value and Rejection Region Based on the information provided, the significance level is α=0.05, the number of degrees of freedom is df = (2 - 1) * (3 - 1) = 2, so the critical value is 5.9915. Then the rejection region for this test becomes R={χ2:χ2>5.9915}. (4)Test Statistics The Chi-Squared statistic is computed as follows: (5)P-value The corresponding p-value for the test is p=Pr(χ2​>1.7958)=0.4074 PART (C) (6)The decision about the null hypothesis Since it is observed that χ2=1.7958<χ2_c​rit=5.9915, it is then concluded that the null hypothesis is NOT rejected. (7)Conclusion It is concluded that the null hypothesis Ho is NOT rejected. Therefore, there is NOT enough evidence to claim that the two variables - Type of planet and Shape of orbit are dependent, at the 0.05 significance level. PART (D) Z test for one population proportion would be most appropriate to test the hypothesis.

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