Question

Please explain this problem step by step, please.

To determine whether there are more “cat people” or “dog people” in the United States, Dr. Rueckert randomly selects 600 US citizens and asks them whether they prefer dogs or cats (they are not allowed to say they like them equally). She finds that 350 people prefer dogs, 250 people prefer cats.

a. What is the null hypothesis?

b. What is the alternative hypothesis?

c. If the null hypothesis is true, how many of the 600 people sampled would you expect to prefer each pet?

d. Calculate the appropriate statistic to test the null hypothesis

e. What are the degrees of freedom?

f. What is the critical value?

g. Should the null hypothesis be rejected? Why, or why not?

Answer 1

Say p0 be the proportion of "dog people" in the US
We will conduct a one-sample proportion z-test.

a) Ho: p0 = 0.5
b) Ha: p0 =/ 0.5
c) If the null hypothesis is true, we would expect 300 people to prefer each pet.
d) The sample statistics are:

p = 350/600 = 0.583

Test statistic is:
z = (p - p0)/√(p0*q0/n)

z = (0.583 - 0.50)/√(0.50*0.50/600)

z = 0.083/0.0204 = 4.07

e) There are no degrees of freedom in a z-statistic.

f) Critical value of z at 0.05 significance level is +-1.96

g) As the z = 4.06 > 1.96, we can reject the null hypothesis and say that the proportion is not 0.5 and there are more dog people in the US.