Question

According to a study done by Wakefield Research, the proportion of Americans who can order a meal in a foreign language is 0.47. Suppose a random sample of 200 Americans is asked to disclose whether they can order a meal in a foreign language. What is the probability the proportion of Americans who can order a meal in a foreign language is greater than 0.5

Answer 1

Probability that the proportion of Americans who can order a meal in a foreign language is greater than 0.5 is 0.19766.

Explanation:

We are given that according to a study done by Wake field Research, the proportion of Americans who can order a meal in a foreign language is 0.47.

Suppose a random sample of 200 Americans is asked to disclose whether they can order a meal in a foreign language.

Let
`\hat p` = sample proportion of Americans who can order a meal in a foreign language

The z-score probability distribution for sample proportion is given by;

Z =
`\frac{ \hat p-p}{\sqrt{(\hat p(1-\hat p))/(n) } }` ~ N(0,1)

where,
`\hat p` = sample proportion

p = population proportion of Americans who can order a meal in a foreign language = 0.47

n = sample of Americans = 200

Probability that the proportion of Americans who can order a meal in a foreign language is greater than 0.5 is given by = P(
`\hat p` > 0.50)

P(
`\hat p` > 0.06) = P(
`\frac{ \hat p-p}{\sqrt{(\hat p(1-\hat p))/(n) } }` >
`\frac{0.5-0.47}{\sqrt{(0.5(1-0.5))/(200) } }` ) = P(Z > 0.85) = 1 - P(Z
`\leq` 0.85)

= 1 - 0.80234 = 0.19766

Now, in the z table the P(Z
`\leq` x) or P(Z < x) is given. So, the above probability is calculated by looking at the value of x = 0.85 in the z table which has an area of 0.80234.

Therefore, probability that the proportion of Americans who can order a meal in a foreign language is greater than 0.5 is 0.19766.