The life span of a calculator manufactured by the Intel Corporation has a normal distribution with a mean of 54 months and a standard deviation of 8 months. The company guarantees that any calculator that starts malfunctioning within 36 months of the purchase will be replaced by a new one. What is the probability that a randomly selected calculator will be replaced by a new one?
http://davidmlane.com/hyperstat/z_table.html
To find the probability that a randomly selected calculator will be replaced by a new one, we can use the concept of the standard normal distribution.
Step 1: Convert the given values into a standard normal distribution
To do this, we use the formula z = (x - μ) / σ, where z is the standard score, x is the value we want to convert, μ is the mean, and σ is the standard deviation. In this case, we want to convert the value 36 months.
z = (36 - 54) / 8 = -18 / 8 = -2.25
Step 2: Find the probability using the standard normal distribution table
Using the standard normal distribution table (also known as the z-table), we can find the probability corresponding to the z-score of -2.25.
Looking at the table, we find that the closest value to -2.25 is -2.28. The corresponding probability is 0.0119.
However, since we're looking for the probability that a calculator will be replaced by a new one (which means it will malfunction within 36 months), we need to find the probability of the calculator lasting longer than 36 months. This is equal to 1 minus the probability of the calculator malfunctioning within 36 months.
P(X > 36) = 1 - P(X ≤ 36)
Since the standard normal distribution is symmetric about 0, we can use the complement rule to find the probability:
P(X > 36) = 1 - P(Z ≤ -2.25)
P(X > 36) = 1 - 0.0119
P(X > 36) ≈ 0.9881
Therefore, the probability that a randomly selected calculator will be replaced by a new one is approximately 0.9881 or 98.81%.