Question

Side Effects for Migraine Medicine

In clinical trials and extended studies of a medication whose purpose is to reduce the pain associated with migraine headaches, 2% of the patients in the study experienced weight gain as a side effect. Suppose a random sample of 600 users of this medication is obtained.

1. Explain why you can use normal approximation to binomial distribution to approximate the probabilities below.

2. Approximate, up to 4 decimal digits, the probability that 20 or fewer users will experience weight gain as a side effect.

3. Approximate, up to 4 decimal digits, the probability that 22 or more users experience weight gain as a side effect.

4. Approximate, up to 4 decimal digits, the probability that between 20 and 30 patients, inclusive will experience weight gain as a side effect.

Answer 1

Solution:-

Given data:-

Sample size, n = 600

p = 0.02

Mean = n*p

where n=600 , p=0.02

µ = 12

Standard deviation = sqrt(npq)

q=1-p=1-0.02

q=0.98

σ = sqrt(600*0.02*0.98)

Standard deviation , σ = 3.4293

(1) Explain why you can use normal approximation to binomial distribution to approximate the probabilities below.

The ordinary dispersion can be utilized as an estimate to the binomial circulation, in specific situations, in particular: If X ~ B(n, p) and if n is huge.

Where X is the random variable which is binomial distribution

As both the conditions are fulfilled we will utilize ordinary estimate to binomial dispersion.

(2)

P(X≤ 20)

P(X ≤ 20+0.5) [from the equation of continuity]

Actually, z=(X - µ)/σ = (20.5-12)/( 3.4293)=2.4787

=P(z ≤ 2.4787)

=0.9934[from z-table]

The probability that 20 or fewer users will experience weight gain as a side effect=0.9934

(3)

P(X ≥ 22)

=P(X ≥ 22-0.5) [from the equation of continuity]

Actually, z=(X - µ)/σ = (21.5-12)/( 3.4293)=2.7703

=1 - P(z < 2.7703)

=0.0028[from z-table]

The probability that 22 or more users experience weight gain as a side effect = 0.0028

(4)

P(20 ≤ X ≤ 30)

P(20-0.5 ≤ X ≤ 30+0.5)[from the equation of continuity]

z1=(X - µ)/σ = (19.5-12)/( 3.4293)=5.3947

z2=(X - µ)/σ = (30.5-12)/( 3.4293)=2.187

= P( 5.3947 ≤ z ≤ 2.187)

= P(z < 5.3947) - P(z < 2.187)=0.0144

Probability that between 20 and 30 patients, inclusive will experience weight gain as a side effect=0.0144