Normal approximation to the binomial

Question 1

When can you use a normal distribution to approximate a binomial distribution?

Answer 1

If n is large enough (n≥30)
If p is roughly 0.5

Also valid if np≥10 and n(1-p)≥10

Question 2

How would you rewrite P(x=5) in terms of y?

Answer 2

p(4.5 ≤ y ≤ 5.5)

Question 3

How would you rewrite P(x≥12.5) in terms of y?

Answer 3

P(y ≥ 11.5)

Question 4

How would you rewrite P(x>12) in terms of y?

Answer 4

P(y > 12.5)

Question 5

How would you rewrite P(x<13) in terms of y?

Answer 5

P(y<12.5)

Question 6

How would you rewrite P(x≤ 13) in terms of y?

Answer 6

P(y≤ 13.5)

Question 7

How would you rewrite P(10 ≤ x < 18) in terms of y?

Answer 7

P(9.5 ≤ y < 17.5)

Question 8

How would you rewrite P(3 < x ≤ 20) in terms of y?

Answer 8

P(3.5 < y ≤ 20.5)

Question 9

X~B (200,0.6)
How would you use the normal distribution to binomial distribution to find P(x<105) ?

Answer 9

change X~B to Y~N by working out μ (np) and σ2 (np(1-p))
So X~B(200,06) changes to Y~N(120,48)
Change P(x<105) to P(Y ≤ 104.5)
then work out P(Y ≤ 104.5) using normal distribution and your μ and σ2 values

Question 10

X~B (200,0.6)
How would you use the normal distribution to binomial distribution to find P(x=122) ?

Answer 10

change X~B to Y~N by working out μ (np) and σ2 (np(1-p))
So X~B(200,06) changes to Y~N(120,48)
Change P(x=122) to P(121.5≤ y ≤ 122.5)
work out P(121.5≤ y ≤ 122.5) using normal distribution and your μ and σ2 values