Question

A random sample is to be selected from a population that has a proportion of successes p.

A. For which of the following sample sizes would the sampling distribution of p̂ be approximately normal if p = 0.75? (select all that apply).

[ ] n = 10

[ ] n = 20

[ ] n = 30

[ ] n = 70

[ ] n = 100

[ ] n = 300​

B. For which of the following sample sizes would the sampling distribution of p̂ be approximately normal if p = 0.4? (Select all that apply.)

[ ] n = 10

[ ] n = 20

[ ] n = 30

[ ] n = 70

[ ] n = 100

[ ] n = 300

Answer 1

(A)

(i) n = 10

Since n not 30, Condition 1 is not satisfied.

So

Sampling Distribution of would not be approximately normal

(ii) n = 20

Since n not 30, Condition 1 is not satisfied.

So

Sampling Distribution of would not be approximately normal

(iii) n = 30

Since n 30, Condition 1 is satisfied.

p = 0.75

So,

np = 30 X 0.75 = 22.5 > 10. So Condition 2 is satisfied.

q = 1 - p = 0.25

So,

nq = 30 X 0.25 = 7.5 not > 10. So, Condition 3 is not satisfied.

So

Sampling Distribution of would not be approximately normal

(iv) n = 70

Since n 30, Condition 1 is satisfied.

p = 0.75

So,

np = 70 X 0.75 = 52.5 > 10. So Condition 2 is satisfied.

q = 1 - p = 0.25

So,

nq = 70 X 0.25 = 17.5 > 10. So, Condition 3 is satisfied.

So

Sampling Distribution of would be approximately normal

(v) n = 100

Since n 30, Condition 1 is satisfied.

p = 0.75

So,

np = 100 X 0.75 = 75 > 10. So Condition 2 is satisfied.

q = 1 - p = 0.25

So,

nq = 100 X 0.25 = 25 > 10. So, Condition 3 is satisfied.

So

Sampling Distribution of would be approximately normal

(vi) n = 300

Since n 30, Condition 1 is satisfied.

p = 0.75

So,

np = 300 X 0.75 = 225 > 10. So Condition 2 is satisfied.

q = 1 - p = 0.25

So,

nq = 300 X 0.25 = 75 > 10. So, Condition 3 is satisfied.

So

Sampling Distribution of would be approximately normal

(B)

(i) n = 10

Since n not 30, Condition 1 is not satisfied.

So

Sampling Distribution of would not be approximately normal

(ii) n = 20

Since n not 30, Condition 1 is not satisfied.

So

Sampling Distribution of would not be approximately normal

(iii) n = 30

Since n 30, Condition 1 is satisfied.

p = 0.4

So,

np = 30 X 0.4 = 12 > 10. So Condition 2 is satisfied.

q = 1 - p = 0.6

So,

nq = 30 X 0.6= 18 > 10. So, Condition 3 is satisfied.

So

Sampling Distribution of would be approximately normal

(iv) n = 70

Since n 30, Condition 1 is satisfied.

p = 0.4

So,

np = 70 X 0.4 = 28 > 10. So Condition 2 is satisfied.

q = 1 - p = 0.6

So,

nq = 70 X 0.6 = 42 > 10. So, Condition 3 is satisfied.

So

Sampling Distribution of would be approximately normal

(v) n = 100

Since n 30, Condition 1 is satisfied.

p = 0.4

So,

np = 100 X 0.4= 40 > 10. So Condition 2 is satisfied.

q = 1 - p = 0.6

So,

nq = 100 X 0.6 = 60 > 10. So, Condition 3 is satisfied.

So

Sampling Distribution of would be approximately normal

(vi) n = 300

Since n 30, Condition 1 is satisfied.

p = 0.4

So,

np = 300 X 0.4 = 120 > 10. So Condition 2 is satisfied.

q = 1 - p = 0.6

So,

nq = 300 X 0.6 = 180 > 10. So, Condition 3 is satisfied.

So

Sampling Distribution of would be approximately normal