Question

The National Geographic Society conducted a study that included 3000 respondents, age 18 to 24 , in nine dif- ferent countries (San Luis Obispo Tribune, November 21 , 2002). The society found that \(10 \%\) of the participants could not identify their own country on a blank world map. a. Construct a \(90 \%\) confidence interval for the proportion who can identify their own country on a blank world map. b. What assumptions are necessary for the confidence interval in Part (a) to be valid? c. To what population would it be reasonable to generalize the confidence interval estimate from Part (a)?

Short answer

a. The 90% confidence interval for the proportion who can identify their own country on a blank world map is approximately (0.89248, 0.90752). \nb. The assumptions necessary for the confidence interval to be valid include that the sample is a simple random sample, and the sampling distribution is approximately normal. \nc. It would be reasonable to generalize this confidence interval to the population of people aged 18 to 24 in the nine countries where the study took place.

Step-by-step solution

Calculate standard error of the proportion

Calculate the standard error (SE) for the proportion using the formula \(\sqrt{P(1-P)/n}\), where P is the sample proportion and n is the sample size. Here, P is \(0.9\) and n is \(3000\). Thus, \nSE = \(\sqrt{(0.9)(1-0.9)/3000} = 0.00457\).

Determine the critical value for a 90% confidence interval

Looking up in a Z-table or using statistical software, find the critical value \(Z*\) for a 90% confidence interval. The critical value for a 90% confidence interval in a standard normal distribution is \(\pm 1.645\).

Calculate the margin of error

Multiply the critical value by the standard error to find the margin of error (E). Here, E = \(Z* \times SE = 1.645 \times 0.00457 = 0.00752\).

Construct the confidence interval

Subtract and add the margin of error from/to the sample proportion to find the lower and upper bounds of the 90% confidence interval. Thus, the confidence interval is \((0.9 - 0.00752, 0.9 + 0.00752) = (0.89248, 0.90752)\).

List the assumptions for the confidence interval

The assumptions needed for the confidence interval to be valid are: 1) The sample is a simple random sample. 2) The sampling distribution of the sample proportion is approximately normal, which is generally satisfied if both \(n \times p\) and \(n \times (1-p)\) are greater than 5. Here, both values are \((3000)(0.9) = 2700\) and \((3000)(1-0.9) = 300\), meaning the assumption is satisfied.

Identify the population this confidence interval applies to

The confidence interval applies to the population from which the sample was drawn. In this case, since the study included respondents aged 18 to 24 in nine different countries, it would be reasonable to generalize this confidence interval to the population of people aged 18 to 24 in those nine countries.