Suppose a sample of 510 new car buyers is drawn. Of those sampled, 142 preferred foreign over domestic cars. Using the data, construct the 85% confidence interval for the population proportion of new car buyers who prefer foreign cars over domestic cars. Round your answers to three decimal places.

Question

Warren Meyers

Mathematics

19 November, 07:47

Suppose a sample of 510 new car buyers is drawn. Of those sampled, 142 preferred foreign over domestic cars. Using the data, construct the 85% confidence interval for the population proportion of new car buyers who prefer foreign cars over domestic cars. Round your answers to three decimal places.

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Kirk · Answer 1

Answer: (0.224, 0.332)

Step-by-step explanation: from the question, the total population is 510 and 142 out of this population prefers foreign cars to domestic ones.

Hence the sample size (n) is 142.

Sample proportion (p) = 142 / 510 = 0.278

q = 1 - 0.278 = 0.722.

We are to construct a 85% confidence interval for sample proportion and this is given by the formulae below.

P = p + Zα/2 * (√ (pq/n) ... For upper limit

P = p - Zα/2 * (√ (pq/n) ... For lower limit

We are using a z test to get our critical value because sample size is greater than 30 (n = 142).

The value of Zα/2 from the standard normal distribution table is 1.44 (this is done at a 15% level of significance).

By substituting the parameters, we have that

For upper limit

P = 0.278 + 1.44 * (√ (0.278*0.722/142)

P = 0.278 + 1.44 (0.0375)

P = 0.278 + 0.054

P = 0.332

For lower limit

P = 0.278 - 1.44 * (√ (0.278*0.722/142)

P = 0.278 - 1.44 (0.0375)

P = 0.278 - 0.054

P = 0.224.

Hence the 85% confidence interval for population proportion is (0.224, 0.332)