Question

Shortly after the major oil spill in the Gulf of Mexico that occurred in April 2010 due to an explosion on an oil rig, a Rasmussan Poll found that 23% of respondents were against offshore drilling for oil. In an effect to confirm these results, and environment group randomly selected 200 adults and asked them their opinions about offshore drilling. A. What is the probability that 40 or more people from this sample are against offshore drilling oil? B. What is the probability that 50 or more people from the sample are against offshore drilling oil? C. If 62 people in the sample indicated that they are against offshore drilling for oil. What conclusions can the environmental group make?

Answer 1

From the information

n = number of adults in sample = 200

P : population Proportion of adults against offshore drilling for oil. = 23% = 0.23

Let the random variable X denotes number of people in a sample against offshore drilling for oil.

The distribution of random variable X is binomial with parameter n = 200 and P = 0.23

X ~ Bin (n = 200, P = 0.23)

The p.m.f. of random variable X is

E(X) = nP = 200 * 0.23 = 46 and SD(X) =sqrt(nP(1-P)) = 5.9515

A) Required probability = P (X > = 40) = P ( X >= 40.5) using continuity correction.

Since n is large, using normal approximation to binomial distribution

Z = (X-E(X)) / SD(X) ~ N(0,1).

= P ( Z >= -0.9241)

From normal probability table

P ( Z>= 0.9241) = 0.8229

P( 40 or more people in the sample are against offshore drilling oil) = 0.8229.

B) Required probability = P (X > = 50) = P ( X >= 50.5) using continuity correction.

= P ( Z >= 0.7561)

From normal probability

P(Z >= 0.7561) = 0.2248

P( 50 or more people in the sample are against offshore drilling oil) = 0.2248.

C) x= number of people in the sample are against offshore drilling oil = 62.

n = number of adults in sample = 200

p: sample proportion of people are against offshore drilling oil

p = x/n = 62 / 200 = 0.31.

Since p > P ( 0.31 > 0.23)

Environmental group conclude that the proportion of people are against offshore drilling oil in a sample is greater than population.