Question

A population of values has a normal distribution with μ = 213.5 and σ = 84.5 .

a) Find the probability that a single randomly selected value is between 219 and 227.9. Round your answer to four decimal places. P(219 < X < 227.9)

b) Find the probability that a randomly selected sample of size n = 233 has a mean between 219 and 227.9. Round your answer to four decimal places. P(219 < M < 227.9)

Answer 1

To find the probability that a single randomly selected value is between 219 and 227.9, we standardize the values using the z-score formula and look up corresponding probabilities in the standard normal distribution table. The probability is approximately 0.0381.

Step-by-step explanation:

To find the probability that a single randomly selected value is between 219 and 227.9, we need to calculate the area under the normal distribution curve between these two values. We can use the z-score formula to standardize the values and then look up the corresponding probabilities in the standard normal distribution table. The z-score for 219 is (219 - 213.5) / 84.5 = 0.065. The z-score for 227.9 is (227.9 - 213.5) / 84.5 = 0.170.

Using the standard normal distribution table, the probability corresponding to a z-score of 0.065 is 0.5260. The probability corresponding to a z-score of 0.170 is 0.5641. Now, to find the probability between these two values, we subtract the probability corresponding to a z-score of 0.065 from the probability corresponding to a z-score of 0.170: 0.5641 - 0.5260 = 0.0381.

Therefore, the probability that a single randomly selected value is between 219 and 227.9 is approximately 0.0381.