Question

Data from a random sample of 2300 adult Americans found that 1300 of those surveyed would prefer to live in a hot climate rather than a cold climate. Do the sample data provide convincing evidence that a majority of all adult Americans prefer a hot climate over a cold climate?

(a) State the null and alternative hypotheses to be tested. (b) Test the relevant hypotheses using α = .01. (Be sure to check all necessary conditions and clearly state your test statistic, P-value, and formal decision.) (c) Write a conclusion statement communicating your findings in a sentence.

Answer 1

A.

The hypthesis to be tested is,

H0 : P = 0.5

(P = proportion of americans preferring hot weather

This mean Americans prefer hot and cold climate equally)

Vs

H1 : P > 0.5

(this means majority of Americans prefer hot climate over cold)

(H0 is null hypthesis and H1 is alternative hypothesis)

B.

P0 value of proportion of Americans prefer hot climate under H0

p^ : sample proportion of Americans prefer hot climate = 1300 / 2300 = 0.5652

Test statistic under H0,

Z = (p^ - P0) /(P0*(1-P0)/n)

Z = 5.9948

The critical value is

Z = Z0.01 = 2.32

As Z > Z we reject H0.

P-value = P(Z > z) = P (Z > 5.9948) = 0

(from statistical table of N(0,1),

value of P(Z > 5) = 0)

P-value < hence we reject H0

C.

Conclusion :

Majority of all adult Americans may prefer hot climate over a cold climate.