Answer :
Final answer:
To find the probability that a single randomly selected value is between 189.3 and 206, use the z-score formula and the standard normal distribution. The probability is approximately 0.1409. To find the probability that a randomly selected sample of size n=208 has a mean between 189.3 and 206, use the Central Limit Theorem to approximate the distribution of the sample mean as a normal distribution. The probability is approximately 0.7622.
Explanation:
To find the probability that a single randomly selected value is between 189.3 and 206, we can use the z-score formula and the standard normal distribution. First, we calculate the z-scores for the lower and upper bounds:
z1 = (189.3 - 211.5) / 99.9 = -0.2222
z2 = (206 - 211.5) / 99.9 = -0.0556
Next, we use a standard normal distribution table or calculator to find the area/probability between these z-scores. The table or calculator gives us:
Area between -0.2222 and -0.0556 = 0.1409
Therefore, the probability that a single randomly selected value is between 189.3 and 206 is approximately 0.1409.
To find the probability that a randomly selected sample of size n=208 has a mean between 189.3 and 206, we can use the Central Limit Theorem and approximate the distribution of the sample mean as a normal distribution. The mean of the sample mean distribution would still be μ=211.5, but the standard deviation would be σ/√n = 99.9/√208 = 6.9631. Using the z-score formula and the standard normal distribution, we calculate the z-scores for the lower and upper bounds:
z1 = (189.3 - 211.5) / 6.9631 = -3.1777
z2 = (206 - 211.5) / 6.9631 = -0.7924
Using a standard normal distribution table or calculator, we find the area/probability between these z-scores:
Area between -3.1777 and -0.7924 = 0.7622
Therefore, the probability that a randomly selected sample of size n=208 has a mean between 189.3 and 206 is approximately 0.7622.
