Answer: Hope it helps!!!
Step-by-step explanation:To solve this problem, we need to standardize the value of X using the formula:
z = (X - μ) / (σ / sqrt(n))
where X is the sample mean, μ is the population mean, σ is the population standard deviation, and n is the sample size.
a) To find the probability that X is less than 95, we first need to standardize the value of 95:
z = (95 - 101) / (20 / sqrt(16)) = -1.6
We can then use a standard normal distribution table or calculator to find the probability:
P(X < 95) = P(z < -1.6) = 0.0548
Therefore, the probability that X is less than 95 is 0.0548 or about 5.48%.
b) To find the probability that X is between 95 and 105, we need to standardize the values of 95 and 105:
z1 = (95 - 101) / (20 / sqrt(16)) = -1.6
z2 = (105 - 101) / (20 / sqrt(16)) = 1.6
We can then use a standard normal distribution table or calculator to find the probability:
P(95 < X < 105) = P(-1.6 < z < 1.6) = 0.8664 - 0.0548 = 0.8116
Therefore, the probability that X is between 95 and 105 is 0.8116 or about 81.16%.
c) To find the value of X such that the probability of X being less than that value is 0.05, we need to use the inverse standard normal distribution:
z = invNorm(0.05) = -1.645
We can then solve for X:
-1.645 = (X - 101) / (20 / sqrt(16))
X - 101 = -1.645 * (20 / sqrt(16))
X = 101 - 2.06
X = 98.94
Therefore, the value of X such that the probability of X being less than that value is 0.05 is 98.94.
d) To find the value of X such that the probability of X being greater than that value is 0.10, we need to use the inverse standard normal distribution:
z = invNorm(0.10) = -1.28
We can then solve for X:
-1.28 = (X - 101) / (20 / sqrt(16))
X - 101 = -1.28 * (20 / sqrt(16))
X = 101 + 1.61
X = 102.61
Therefore, the value of X such that the probability of X being greater than that value is 0.10 is 102.61.