If 𝑋 is a normal random variable with mean of 10 and standard deviation of 2.5 such that the probability that 𝑋 is greater than or equal to some constant 𝑘 is 0.1056, find 𝑘.
Let’s think about what we know. We have a mean of 10, a standard deviation of 2.5, and a normal distribution. We can say that the variable 𝑋 is normally distributed with a mean of 10 and a variance of 2.5 squared. The variance is equal to its standard deviation squared. We also know the probability that 𝑋 is greater than or equal to some constant 𝑘 must be 0.1056.
We know that a normal distribution is a bell-shaped curve that is symmetrical about the mean. That is to say the area under the curve on the right is equal to the area under the curve on the left. And these areas represent probability. The area below the full curve is one. The area to the left of this 𝑋-value represents the probability that 𝑋 is less than or equal to whatever this value is. Our variable is 𝑘. And we know the probability that 𝑋 is greater than or equal to 𝑘. We know the area to the right of 𝑘 is equal to 0.1056.
Now, we need to remember what we said about this curve being symmetrical around the mean. This means that we know the area of the values to the right of the mean must be equal to 0.5 as is the area to the left of the mean. And that means to find the area between the mean and 𝑘, we would need to subtract 0.1056 from 0.5 like this, which is equal to 0.3944. At this point, we’ll use a table of areas underneath the standard normal distribution curve. It’s sometimes called a 𝑍-score chart. The table we’re given is in the format the probability that 𝑍 falls between zero and another value lowercase 𝑧.
Since we know that the probability that our 𝑋-value is between the mean and some value 𝑘, we look on the chart for 0.3944. This tells us that our 𝑧-value must be 1.25. If our 𝑍-score is 0.3944, 𝑧 equals 1.25. We need to be really careful here. This value 1.25 is a 𝑧-score. And we need to solve this formula for our 𝑘 value. We found our 𝑧-score 1.25. And that will be equal to 𝑘 minus our mean which is 10 divided by our standard deviation of 2.5. So we multiply both sides by 2.5. 2.5 times 1.25 equals 3.125 which equals 𝑘 minus 10. We then add 10 to both sides and 𝑘 equals 13.125.
This 𝑧-score 1.25 tells us how many standard deviations 𝑘 is from the mean. If 𝑘 is one full standard deviation away from the mean plus one-fourth standard deviation away from the mean, it is 3.125 away from the mean. The mean was 10. 10 plus 3.125 equals 13.125.