Video Transcript
A crop of apples has a mean weight
of 105 grams and a standard deviation of three grams. It is assumed that a normal
distribution is an appropriate model for this data. What is the approximate probability
that a randomly selected apple from the crop has a weight between 102 grams and 108
grams?
Remember, the graph of the curve
representing the normal distribution with a mean of 𝜇 is bell-shaped and symmetric
about that mean, and the total area under the curve is 100 percent or one. The mean weight of the crop of the
apples is 105 grams, and the standard deviation — that’s the measure of spurt here —
is three grams.
The question’s asking us to
calculate the probability that a randomly selected apple has a weight between 102
grams and 108 grams. That’s represented by the area
shaded. Once we’ve established this, our
next step is to calculate the 𝑍-value. This is a way of scaling the data
or standardizing it, in what becomes a standard normal distribution. Once we’ve completed this step, we
can work from a single standard normal table.
We’ll begin by looking at the value
of 108 grams. We have a mean, 𝜇, of 105 and a
standard deviation, 𝜎, of three. So our 𝑍-value here is found by
subtracting 105 from 108 and then dividing by three. That gives us a 𝑍-value of
one. We can then look up a 𝑍-value of
one in our standard normal table. That will tell us the probability
that 𝑍 is less than one, which in turn tells us the probability that 𝑋 is less
than 108: a randomly selected apple has a weight of less than 108 grams.
Looking this value up in our table
and we can see that the probability that 𝑍 is less than one is 0.8413 and the
probability that a randomly selected apple has a weight of less than 108 grams is
0.8413. If we look back to our curve
though, that’s everything to the left of 108.
To find the probability that the
apple has a weight between 102 and 108 grams, we’ll subtract the probability of it
being less than 102 grams from the probability of it being less than 108 grams. That will go back and give us the
area shaded.
Let’s look at an 𝑋-value of 102
then. 𝑍 is equal to 102 minus 105 all
over three, which is negative one. We can’t look up a negative
𝑍-value though, so we use the symmetry of the curve to help us. Since the curve is symmetrical
about the mean, and when we standardize and find the 𝑍-value — we have a mean of
zero — we know the probability that 𝑍 is less than negative one must be the same as
the probability that 𝑍 is greater than one.
Earlier, we said that the area
under the curve is one whole, so we can subtract the probability that 𝑍 is less
than one from one whole. And that will tell us the
probability that 𝑍 is greater than one. One minus 0.8413 is equal to
0.1587. And therefore, the probability that
𝑍 is less than negative one is 0.1587. And in turn, the probability that
𝑋 is less than 102 is 0.1587. Subtracting the probability that 𝑋
is less than 102 from the probability that the randomly selected apple has a weight
of less than 108 gives us 0.6826.
To get our answer as a percentage,
we multiply it by 100. As a percent to the nearest whole
number, the probability that a randomly selected apple has a weight between 102
grams and 108 grams is 68 percent.
