Question

Applying the Concepts 15. Foreign Language According to a study done by Wakefield Research, the proportion of Americans who can order a meal in a foreign language is 0.47. (a) Suppose a random sample of 200 Americans is asked to disclose whether they can order a meal in a foreign language. Is the response to this question qualitative or quantitative? Explain. (b) Explain why the sample proportion, ſ, is a random variable. What is the source of the variability? (c) Describe the sampling distribution of Ô, the proportion of Americans who can order a meal in a foreign language. Be sure to verify the model requirements. (d) In the sample obtained in part (a), what is the probability the proportion of Americans who can order a meal in a foreign language is greater than 0.5? (e) Would it be unusual that, in a survey of 200 Americans, 80 or fewer Americans can order a meal in a foreign language? Why?

Answer 1

Answer:

Given,

p = 0.47

sample n = 200

a)

Here the response of the question is qualitative.

b)

np >= 200*0.47 = 94 > 10

np(1-p) = 200*0.47(1-0.47) = 49.82 >= 10

Here p^ is approximately normal.

c)

Mean = p = 0.47

Standard deviation = sqrt(pq/n) = sqrt(0.47*0.53/200) = 0.0353

d)

P(p > 0.5) = P(z > (0.5 - 0.47)/0.0353)

= P(z > 0.85)

= 0.1976625 [since from z table]

= 0.1977

e)

P(p <= 80) = P(p^ <= 80/200)

= P(p^ <= 0.4)

= P(z < (0.4 - 0.47)/0.0353)

= P(z < - 1.98)

= 0.0238518 [since from z table]

= 0.0239