Suppose 𝑋 is normally distributed
with mean 𝜇 and variance 196. Given that the probability 𝑋 is
less than or equal to 40 is 0.0668, find the value of 𝜇.
We are told that 𝑋 is normally
distributed with a mean 𝜇, which is what we’re looking to find, and a variance. That’s a 𝛿 squared of 196. We’re also told that the
probability that 𝑋 is less than or equal to 40 is equal to 0.0668.
Now, our first step is usually to
find associated 𝑍-scores. So in this example, if we were
looking to find the probability that 𝑋 is less than or equal to 40, we would find
the 𝑍-score associated with an 𝑋-value of 40. However, we don’t yet know the
value of the mean. So we’re going to need to find the
value of 𝑍 such that the probability that 𝑍 is less than or equal to lowercase 𝑧
is equal to 0.0668. Now, what we might do is look for
this probability in the standard normal table.
However, if you try that, you’ll
notice that the smallest possible value we get is 0.5. And that’s because the probability
0.0668 is such a small value it will lie all the way down here. So we need to find a 𝑍-value for
which this is true.
And to do that, we recall that the
standard normal curve is completely symmetrical about the mean. And the area under the curve is one
whole. So this part up here will be
0.0688. And since the probabilities are
cumulative, to find the 𝑍-value for which this is true, we’re going to subtract
0.0688 from one since that’s the total area under the curve. And that’s 0.9332.
Let’s find this value in the
standard normal table. In fact, the 𝑍-value for which
this is true is 1.5. And using the symmetry of our
normal curve, we can see that the probability that 𝑍 is less than or equal to
negative 1.5 is 0.0688. And that’s the value we’re looking
And now, we’re almost ready to
substitute what we know into the formula for the 𝑍-value. Remember, this formula requires the
standard deviation, not the variance. So we find the square root of
196. And that gives us 14. The standard deviation has to be
positive. So we disregard any negative values
for the square root of 196. And now, we can substitute what we
know about 𝑋 into this formula. We have a 𝑍-value of negative
1.5. And that corresponds to an 𝑋-value
of 40. And we’ve just shown that the
standard deviation is 14.
We’re going to solve this equation
for 𝜇. And we do that by multiplying both
sides by 14. That gives us negative 21 equals 40
minus 𝜇. There are a number of different
next steps we can take. One way is to multiply through by
negative one. Or we can add 𝜇 to both sides. Adding 𝜇 gives us 𝜇 minus 21 is
equal to 40. And finally, to solve for 𝜇, we
add 21 to both sides. 40 plus 21 is 61.
And we can now see that the value
of 𝜇, the mean, is 61.