Current Attempt in Progress
A population proportion is 0.59 . Suppose a random sample of 663 items is sampled randomly from this population.
AppendixAS
a. What is the probability that the sample proportion is greater than 0.62 ?
b. What is the probability that the sample proportion is between 0.56 and 0.62 ?
c. What is the probability that the sample proportion is greater than 0.58 ?
d. What is the probability that the sample proportion is between 0.55 and 0.56 ?
e. What is the probability that the sample proportion is less than 0.50 ?

Step 1 ：Given that the population proportion (p) is 0.59 and the sample size (n) is 663.

Step 2 ：We can use the normal approximation to the binomial distribution to solve this problem. The mean (μ) of the sampling distribution of the sample proportion is equal to the population proportion (p), and the standard deviation (σ) is given by the formula \(\sqrt{p(1-p)/n}\).

Step 3 ：So, \(\mu = p = 0.59\)

Step 4 ：And \(\sigma = \sqrt{p(1-p)/n} = \sqrt{0.59(1-0.59)/663} = 0.019\)

Step 5 ：a. What is the probability that the sample proportion is greater than 0.62?

Step 6 ：We first convert 0.62 to a z-score using the formula z = (x - μ)/σ.

Step 7 ：\(z = (0.62 - 0.59)/0.019 = 1.58\)

Step 8 ：The probability that the z-score is greater than 1.58 is \(\boxed{0.0571}\) (from z-table).

Step 9 ：b. What is the probability that the sample proportion is between 0.56 and 0.62?

Step 10 ：We convert 0.56 and 0.62 to z-scores.

Step 11 ：\(z1 = (0.56 - 0.59)/0.019 = -1.58\)

Step 12 ：\(z2 = (0.62 - 0.59)/0.019 = 1.58\)

Step 13 ：The probability that the z-score is between -1.58 and 1.58 is \(\boxed{0.8859}\) (from z-table).

Step 14 ：c. What is the probability that the sample proportion is greater than 0.58?

Step 15 ：We convert 0.58 to a z-score.

Step 16 ：\(z = (0.58 - 0.59)/0.019 = -0.53\)

Step 17 ：The probability that the z-score is greater than -0.53 is \(\boxed{0.7023}\) (from z-table).

Step 18 ：d. What is the probability that the sample proportion is between 0.55 and 0.56?

Step 19 ：We convert 0.55 and 0.56 to z-scores.

Step 20 ：\(z1 = (0.55 - 0.59)/0.019 = -2.11\)

Step 21 ：\(z2 = (0.56 - 0.59)/0.019 = -1.58\)

Step 22 ：The probability that the z-score is between -2.11 and -1.58 is \(\boxed{0.0279}\) (from z-table).

Step 23 ：e. What is the probability that the sample proportion is less than 0.50?

Step 24 ：We convert 0.50 to a z-score.

Step 25 ：\(z = (0.50 - 0.59)/0.019 = -4.74\)

Step 26 ：The probability that the z-score is less than -4.74 is almost \(\boxed{0}\) (from z-table).