A random variable is normally distributed with a mean of 50 and a standard deviation of 5.
b. What is the probability that the random variable will assume a value between 45 and 55 (to 4 decimals)?
c. What is the probability that the random variable will assume a value between 40 and 60 (to 4 decimals)?
To find the probability of a random variable assuming a particular range of values, we need to use the properties of the normal distribution. We will use the standard normal distribution (mean = 0, standard deviation = 1) and then convert it to the given distribution using the formula:
Z = (X - μ) / σ
Where:
Z is the standardized variable (standard normal distribution)
X is the value of the random variable
μ is the mean of the distribution
σ is the standard deviation of the distribution
b. To find the probability that the random variable will assume a value between 45 and 55:
Step 1: Standardize the values:
Z1 = (45 - 50) / 5 = -1
Z2 = (55 - 50) / 5 = 1
Step 2: Look up the standardized values in the standard normal distribution table or use a calculator to find the corresponding probabilities.
Using a standard normal distribution table, the probability for Z between -1 and 1 is approximately 0.6827.
c. To find the probability that the random variable will assume a value between 40 and 60:
Step 1: Standardize the values using the same formula as above:
Z1 = (40 - 50) / 5 = -2
Z2 = (60 - 50) / 5 = 2
Step 2: Look up the standardized values in the standard normal distribution table or use a calculator to find the corresponding probabilities.
Using a standard normal distribution table, the probability for Z between -2 and 2 is approximately 0.9545.
Therefore, for b, the probability is approximately 0.6827, and for c, the probability is approximately 0.9545.