Step 1 :Given that the population proportion is 0.59 and a random sample of 660 items is taken from this population.
Step 2 :We are asked to find the probability that the sample proportion is greater than 0.62.
Step 3 :This is a problem of normal approximation to the binomial distribution. The sample proportion follows a normal distribution with mean equal to the population proportion (0.59) and standard deviation equal to the square root of \((0.59)(1-0.59)/660\).
Step 4 :We need to calculate the z-score for 0.62, which is \((0.62-0.59)\) divided by the standard deviation.
Step 5 :Then we can use the standard normal distribution to find the probability that the z-score is greater than the calculated value.
Step 6 :The calculated z-score is approximately 1.57.
Step 7 :The probability that the sample proportion is greater than 0.62 is approximately 0.0586.
Step 8 :Final Answer: The probability that the sample proportion is greater than 0.62 is \(\boxed{0.0586}\).