A random sample of size n = 270 is taken from a population with a population proportion p = 0.58. [You may find it useful to reference the z table.]

Question

a. Calculate the expected value and the standard error for the sampling distribution of the sample proportion. (Round "expected value" to 2 decimal places and "standard error" to 4 decimal places.)



b. What is the probability that the sample proportion is between 0.50 and 0.60? (Round final answer to 4 decimal places.)



c. What is the probability that the sample proportion is less than 0.50? (Round final answer to 4 decimal places.)

Answer 1

a. The expected value of the sample proportion is equal to the population proportion, so expected value = p = 0.58.

The standard error is calculated using the formula:

standard error = sqrt((p(1-p))/n)

Substituting the values in, we get:

standard error = sqrt((0.58(1-0.58))/270) ≈ 0.0336

b. To find the probability that the sample proportion is between 0.50 and 0.60, we need to calculate the z-scores for these values and then use the z-table to find the respective probabilities.

The z-score for 0.50 is calculated using the formula:

z = (x - μ) / σ

Substituting the values in, we get:

z1 = (0.50 - 0.58) / 0.0336 ≈ -2.38

Similarly, the z-score for 0.60 is:

z2 = (0.60 - 0.58) / 0.0336 ≈ 0.60

Using the z-table, we find the corresponding probabilities:

P(0.50 < x < 0.60) ≈ P(-2.38 < z < 0.60) ≈ P(z < 0.60) - P(z < -2.38)

Using the z-table, we find the respective probabilities:

P(z < 0.60) ≈ 0.7257
P(z < -2.38) ≈ 0.0088

P(0.50 < x < 0.60) ≈ 0.7257 - 0.0088 = 0.7169

So, the probability that the sample proportion is between 0.50 and 0.60 is approximately 0.7169.

c. To find the probability that the sample proportion is less than 0.50, we use the z-score formula and the z-table.

The z-score for 0.50 is:

z = (x - μ) / σ

Substituting the values in, we get:

z = (0.50 - 0.58) / 0.0336 ≈ -2.38

Using the z-table, we find the probability:

P(x < 0.50) ≈ P(z < -2.38) ≈ 0.0088

So, the probability that the sample proportion is less than 0.50 is approximately 0.0088.