Video Transcript
Let 𝑋 be a random variable which
is normally distributed with mean 𝜇 equals 75 and standard deviation 𝜎 equals
six. Given that the probability 𝑋 is
greater than or equal to 𝑘 equals 0.9938, find 𝑘.
We’re told that 𝑋 is a normal
random variable with a mean of 75 and a standard deviation of six. We’re also given the probability
that this normal random variable 𝑋 is greater than some unknown value 𝑘 is
0.9938. We can visualize this probability
as the area to the right of the value of 𝑘 under the normal distribution curve. As this probability is greater than
0.5, we know that 𝑘 is in the lower half of the distribution, as the normal
distribution is symmetrical about its mean. So the probability either side of
the mean is 0.5.
We can deduce then that if the
probability to the right of the mean is 0.5, then the probability that 𝑋 is greater
than or equal to 𝑘 and less than or equal to the mean of 75 is 0.9938 minus 0.5,
which is 0.4938. Now, we’ll need to use statistical
tables later. But in order to find probabilities
for the normal distribution, we need to first standardize the distribution. That is, we need to convert to a
normal distribution with a mean of zero and a standard deviation of one.
We can convert any observation 𝑋
from any normal distribution with a mean of 𝜇 and a standard deviation 𝜎 to a
standardized scale using the formula 𝑧 equals 𝑥 minus 𝜇 over 𝜎. And these standardized values are
known as 𝑧-scores. For the observation 𝑘 from a
normal distribution with a mean of 75 and a standard deviation of six, the 𝑧-score
is 𝑘 minus 75 over six. By standardizing, we find that the
probability 𝑋 is greater or equal to 𝑘 and less than or equal to 75 is the same as
the probability that the standard normal random variable 𝑍 is greater than or equal
to 𝑘 minus 75 over six and less than or equal to zero. And we know this probability is
0.4938.
We’d like to use our statistical
tables to now work out the value of 𝑘. But the tables give probabilities
in a specific form. They give the probability that 𝑍
is between zero and a positive 𝑧-score. This is where we need to use the
symmetry of the normal distribution. The normal distribution is
symmetrical about its mean. So the probability that 𝑍 is
between some negative value and zero is the same as the probability that 𝑍 is
between zero and the absolute value of that negative value. So the probability that 𝑍 is
between 𝑘 minus 75 over six and zero is the same as the probability that 𝑍 is
between zero and negative 𝑘 minus 75 over six. This probability we know to be
equal to 0.4938. So we can use our statistical
tables.
Now, you may need to zoom in to see
this. But if you do, you’ll see that a
probability of 0.4938 is associated with a 𝑧-score of 2.5. This tells us then that negative 𝑘
minus 75 over six is equal to 2.5. And we can solve this equation to
find the value of 𝑘. Multiplying both sides by negative
one, we have 𝑘 minus 75 over six equals negative 2.5. We then multiply each side of the
equation by six to give 𝑘 minus 75 equals negative 15. And finally we add 75 to each side
to give 𝑘 is equal to 60.
This answer makes sense because 60
is less than 75. So the value of 𝑘 is in the lower
half of the distribution. And we can be confident that the
work we did in reflecting this region was correct. So by standardizing the normal
distribution and then reflecting an area in the lower half of the distribution to
become an area in the upper half of the distribution, we were able to use our
statistical tables and then rearrange the equation this gave to find that the value
of 𝑘 is 60.
