# Video: Estimating Population Percentages from a Normal Distribution in Context

The marks from a statistics exam are normally distributed with mean π and standard deviation π. What percentage of students got a mark between (π β 2.27 Ο) and (π + 1.73 Ο)?

04:09

### Video Transcript

The marks from a statistics exam are normally distributed with mean π and standard deviation π. What percentage of students got a mark between π minus 2.27π and π plus 1.73π.

Remember, the graph of a normally distributed data set is this bell-shaped curve. Itβs completely symmetrical about the mean and the area underneath the curve is one or 100 percent. In this question, weβre looking to find the percentage of students who got a mark between π minus 2.27π and π plus 1.73π. And to do this, we need to find the associated π§-scores. This is essentially a way of standardizing our data. And it allows us to read results of a standard normal table with a mean of zero and a standard deviation of one.

Now at this point, we shouldnβt worry too much that we donβt have numerical values in the question. Instead, weβre going to go ahead and substitute what we know about our data set into the formula for the π§-score. The lower end of our interval is π minus 2.27π. So, weβre going to make this our first π₯-value. And this means our first π§-value is found by π minus 2.27π minus π all over π. π minus π is zero. And we can also divide through by π. And we can see that our lower π§-score is negative 2.27.

Letβs repeat this process with the π₯-value at the upper end of our interval. Itβs π plus 1.73π. So, our associated π§-value here is π plus 1.73π minus π all over π. Once again, π minus π is zero. And we divide through by π. And we can see that this π§-score is 1.73. So, we can see that to find the percentage of students who got a mark between π minus 2.27π and π plus 1.73π, we need to find the probability that π§ is greater than negative 2.27 and less than 1.73.

And in fact, with normal distribution, the probabilities are cumulative. So, if we find the probability that π§ is less than 1.73, thatβs the same as the probability that π₯ is less than π plus 1.73π. Thatβs everything to the left-hand side.

To find the shaded area, weβre going to need to subtract the probability that π is less than 2.27π, or the probability that π§ is less than negative 2.27. And we can find the probability that π§ is less than 1.73 by looking the value of 1.73 up in the standard normal table.

But youβll notice that we canβt find negative 2.27 in that table. Instead, we need to use the symmetry of the curve. And we can see that the probability that π§ is less than negative 2.27 is equal to the probability that π§ is greater than 2.27. So, how do we find the probability that π§ is greater than 2.27?

Well, remember we said that the probabilities are cumulative. And, earlier, we said that they also added to one. So, we can subtract the probability that π§ is less than 2.27 from one. And that will give us the probability that π§ is greater than 2.27 and, in turn, the probability that π§ is less than negative 2.27.

If we find a π§-score of 2.27, we see it has an associated probability of 0.9884. And 0.9582 minus one minus 0.9884 is equal to 0.9466. Weβre being told to find the percentage of students that got a mark between π minus 2.27 and 1.73π. To change from a decimal to a percentage, we multiply our number by 100. 0.9466 multiplied by 100 is 94.66. And our answer is 94.66 percent.