Sample Proportions Exercises

1. The proportion of a population with a characteristic of interest is p = 0.37. Random samples of size 1,600 are taken.

a. Is normal approximation valid for this situation? Explain.

• Yes, np ≥ 10 and n(1 - p) ≥ 10

• No, np < 10

• No, n( 1 - p) < 10

• No, np < 10 and n(1 - p) < 10

1. The proportion of a population with a characteristic of interest is p = 0.37. Random samples of size 1,600 are taken.

b. Find the mean and standard deviation of the sample proportion.

• $\mu_{\rho}=\ .37\ and\ \sigma_{\rho}=.0121$

• $\mu_{\rho}=\ .0121\ and\ \sigma_{\rho}=.37$

• $\mu_{\rho}=\ .37\ and\ \sigma_{\rho}=2.3951$

• $\mu_{\rho}=\ .37\ and\ \sigma_{\rho}=1.4569$

2. The proportion of a population with a characteristic of interest is p = 0.37. Random samples of size 125 are taken.

a. Is normal approximation valid for this situation? Explain.

• Yes, np ≥ 10 and n(1 - p) ≥ 10

• No, np < 10

• No, n(1 - p) < 10

• No, np < 10 and n(1 - p) < 10

2. The proportion of a population with a characteristic of interest is p = 0.37. Random samples of size 125 are taken.

b. Find the mean and standard deviation of the sample proportion.

• $\mu_{\rho}=46.25\ and\ \sigma_{\rho}=.0432$

• $\mu_{\rho}=.0432\ and\ \sigma_{\rho}=46.25$

• $\mu_{\rho}=.37\ and\ \sigma_{\rho}=.0019$

• $\mu_{\rho}=.37\ and\ \sigma_{\rho}=.0432$

3. A random sample of size 121 is taken from a population in which the proportion with the characteristic of interest is p = 0.47. Find the indicated probabilities:

a. Probability that the average is between 0.45 and 0.50.

• ncdf(0.45, 0.50, 0.47, 0.0454)

• ncdf(0.50,0.45, 0.47, 0.0454)

• ncdf(-1000, .50, .47, 0.454)

• ncdf(0.45, 1000, .47, 0.454)

3. A random sample of size 121 is taken from a population in which the proportion with the characteristic of interest is p = 0.47. Find the indicated probabilities:

b. Probability that the average is greater than or equal to 0.50.

• ncdf(0.45, 0.50, 0.47, 0.0454)

• ncdf(0.50,0.45, 0.47, 0.0454)

• ncdf(-1000, .50, .47, 0.454)

• ncdf(0.5, 1000, .47, 0.454)

4. An airline claims that 72% of all its flights to a certain region arrive on time. In a random sample of 36 recent arrivals, 23 were on time.

a. Compute the sample proportion.

• 0.6389

• 1.5652

• 25.92

• 16.56

4. An airline claims that 72% of all its flights to a certain region arrive on time. In a random sample of 36 recent arrivals, 23 were on time.

b. Assuming the airline’s claim is true, find the probability of a sample of size 36 producing the sample proportion you got in part (a) or lower using normal approximation.

• ncdf(-1000, 36, 0.72, 0.0748)

• ncdf ( -1000, 0.6389, .72, 0.0748)

• ncdf (0.6389, 1000, 0.72, 0.0748)

• ncdf (0.6389, 1000, 0.0748, 0.72)

5. An economist investigates whether people are keeping cars longer now than in the past. He knows that five years ago, 38% of all passenger vehicles in operation were at least 10 years old. A study is done in which 325 automobiles are randomly sampled. Of them, 132 are ten years or older.

a. Find the sample proportion.

• 0.0308

• 2.4621

• 0.4062

• 0.0758

5. An economist investigates whether people are keeping cars longer now than in the past. He knows that five years ago, 38% of all passenger vehicles in operation were at least 10 years old. A study is done in which 325 automobiles are randomly sampled. Of them, 132 are ten years or older.

b. Find the probability that, when a sample of size 325 is drawn from a population in which the true proportion is 0.38, you will get the proportion you got in part (a) or larger using normal approximation.

• ncdf(0.4062, 1000, 0.38, 0.0269)

• ncdf(0.38, 1000, 0.4062, 0.0269)

• ncdf( -1000, 0.38, 0.4062, 0.0269)

• ncdf(-1000, 0.4062, 0.38, 0.0269)