Question

A claim is made that at least 10% of the students at the university are from out of state. To test this, 80 students in a random sampling were asked whether they came from out of state with the result that 6 reported that they did. Try to disprove this claim at the .05 level.

Answer 1

Let P : proportion of out of state students.

The claim is that atleast 10% students are out of state, to disprove this we need to prove that they are less than 10%

We have to test the hypothesis :

H0 : P >= 0.10

V/s

H1 : P < 0.10

Or we can write

H0 : P = 0.10

(as we need atleast 10% of students who are out of state to prove the claim)

V/s

H1 : P < 0.10

The test statistic un der H0,

Z = (p^ - P0) /{P0*(1-P0)/n} which follows N(0,1)

P0 : value of P under H0

p^ : sample proportion = 6/80

n=80

Hence,

Z = - 0.7454

Z = Z0.05 = 1.64

Here Z > - Z, hence we accept H0 at 5 % level of significance.

Conclusion :

The claim that atleast 10% students are from out of state may be right.