Question

A test of hypothesis was conducted to determine if the population proportion of all college students that had a tattoo was less than 30%. A printout for this analysis appears below.

Hypothesis Test - One Proportion

Sample Size 100
Successes 23
Proportion 0.23000

Null Hypothesis: P = 0.3
Alternative Hyp: P < 0.3

Difference -0.07000
Standard Error 0.04208
Z (uncorrected) -1.53 P 0.0633

Based on the information given, would the sample size be considered a large sample?

Yes. All sample sizes are considered large when working with proportions.

Yes, since n > 30

No. It would not be considered a large sample.

Yes, since np > 15 and nq > 15.

Answer 1

To Test :-

H0 :- P = 0.3

H1 :- P < 0.3

P0 = 0.3
q0 = 1 - P0 = 0.7
n = 100
P = X / n = 23/100 = 0.23

Test Statistic :-
Z = ( P - P0) / √(P0 * q0 / n)
Z = ( 0.23 - 0.3 ) / √(( 0.3 * 0.7) /100)
Z = -1.5275

Test Criteria :-
Reject null hypothesis if Z < -Z(α)
Z(α) = Z(0.063) = 1.5301
Z > -Z(α) = -1.5275 > -1.5301, hence we fail to reject the null hypothesis
Conclusion :- We Fail to Reject H0

Decision based on P value
P value = P ( Z < -1.5275 )
P value = 0.0633
Reject null hypothesis if P value < α = 0.063
Since P value = 0.0633 > 0.063, hence we fail to reject the null hypothesis
Conclusion :- We Fail to Reject H0

Based on the information given, would the sample size be considered a large sample?

np = 100 * 0.23 = 23 > 15

nq = 100 * ( 1 - 0.23 ) = 77 > 15

Yes, since np > 15 and nq > 15.