Question

Seventy percent of adults favor some kind of government control on the prices of medicines. A random sample of 400 adults was selected to determine whether or not they favor some kind of government control.

(a) Are the conditions necessary to apply the CLT satisfied? Explain.

(b)Describe and sketch the sampling distribution of the sample proportion in this situation.

(c) Find the probability that the proportion of adults who favor some kind of government control is less than 0.65.

(d)Find the probability that the proportion is between 0.73 and 0.76.

Answer 1

We have given p = 0.70 and n = 400

a) n*p = 400*0.70 = 280

n*(1-p) = 400*(1 - 0.70) = 120

Since np and n(1-p) are both greater than 10 so condition for CLT are satisfied.

b) The sampling distribution of sample proportion will be normal distribution.

Mean is

Standard deviation is

The curve is

c) We have to find P( < 0.65)

For finding this probability we have to find z score.

That is we have to find P(Z < - 2.18)

P(Z < - 2.18) = 0.0145 ( Using z table)

So the probability that the proportion of adults who favor some kind of government control is less than 0.65 is 0.0145

d) We have to find the probability that the proportion is between 0.73 and 0.76.

That is we have to find P( 0.73 <   < 0.76)

For finding this probability we have to find z score.

That is we have to find P(1.31 < Z < 2.62)

P(1.31 < Z < 2.62) = P(Z < 2.62) - P(Z < 1.31) = 0.9956 - 0.9049 = 0.0907

So the probability that the proportion is between 0.73 and 0.76 is 0.0907