Final answer:
To find the probability that a random sample has a mean between two values, we can use the fact that the distribution of sample means for a normal population follows the normal distribution. For a random sample of 4, the probability is 0.4918. For a random sample of 14, the probability is 0.1042. For a random sample of 25, the probability is 0.4918.
Step-by-step explanation:
To find the probability that a random sample has a mean between two values, we can use the fact that the distribution of sample means for a normal population follows the normal distribution.
a. For a random sample of 4, we can calculate the z-scores for both values and then find the probability of the sample mean falling between these z-scores by using the standard normal distribution table or a calculator. The z-scores are (347-340)/(32/sqrt(4)) = 1 and (351-340)/(32/sqrt(4)) = 2.5. Using the standard normal distribution table, the probability is P(1 < Z < 2.5) = 0.4918.
b. For a random sample of 14, we can follow the same steps as in part a to calculate the z-scores and find the probability. The z-scores are (347-340)/(32/sqrt(14)) = 0.8292 and (351-340)/(32/sqrt(14)) = 1.5638. Using the standard normal distribution table, the probability is P(0.8292 < Z < 1.5638) = 0.1042.
c. For a random sample of 25, we can again calculate the z-scores and find the probability. The z-scores are (347-340)/(32/sqrt(25)) = 1.25 and (351-340)/(32/sqrt(25)) = 2.5. Using the standard normal distribution table, the probability is P(1.25 < Z < 2.5) = 0.4918.