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

In a school, the weights of students are normally distributed with mean 66 kg and variance 16 kgΒ². What percentage of students weigh between 54 kg and 70 kg?

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Video Transcript

In a school, the weights of students are normally distributed with mean 66 kilograms and variance 16 kilograms squared. What percentage of students weigh between 54 kilograms and 70 kilograms?

We have been told that this data set is normally distributed. And it has a mean πœ‡ of 66 and a variance that’s 𝜎 squared of 16. We’re looking to find the percentage of students that weigh between 54 kilograms and 70 kilograms.

And it can be useful to consider that bell-shaped curve that describes a normal distribution. It’s completely symmetrical about the mean and the total area under the curve is one or 100 percent. Remember the mean for our data set is 66. And since we’re looking to find the percentage of students that weigh between 54 and 70 kilograms, we need to find the probability that a student falls in the shaded area.

And to do that, we need to consider standardizing our data using the 𝑍 formula. This is a way of scaling our information. And it allows us to find probabilities from the standard normal table. That has a mean πœ‡ of zero and a standard deviation of one.

The formula for the 𝑍 score is 𝑋 minus πœ‡ β€” that’s the mean β€” over 𝜎, where 𝜎 is the standard deviation. We can find the square root of the variance. And doing so, we see that the standard deviation of our set of data is four. The two 𝑋-values that we’re looking to find corresponding 𝑍 values for are 54 kilograms and 70 kilograms.

For an 𝑋-value of 54, our 𝑍 formula is 54 minus 66 over four. That’s negative three. And for an 𝑋-value of 70, our formula becomes 70 minus 66 over four. And we can see that our second 𝑍 score is one. So we can see that to find the probability that a student weighs between 54 and 70 kilograms, we need to find the probability that 𝑍 is greater than negative three and less than one.

And since these probabilities are cumulative, we can say that to find this probability, we’re going to subtract the probability that 𝑍 is less than or equal to negative three from the probability that 𝑍 is less than one. And we can look up a 𝑍 value of one in our standard normal table, it’s 0.8413. So the probability that 𝑍 is less than one or the probability that a student weighs less than 70 kilograms is 0.8413.

But what about the probability that 𝑍 is less than negative three? There are no negative values in our standard normal table. Well, if we look at our standard normal curve, we can see that the probability that 𝑍 is less than negative three is this shaded region.

And we said that this curve is completely symmetrical about the mean. So we can say that the probability that 𝑍 is less than negative three is the same as the probability of that 𝑍 is greater than three. And in fact to find the probability that 𝑍 is greater than three, we can say we’ll subtract the probability that 𝑍 is less than three from one since the total area under the curve is one. The probability that 𝑍 is less than three according to our standard normal table is 0.9987.

So the probability that 𝑍 is greater than three or the probability that 𝑍 is less than negative three is one minus 0.9987, which is 0.0013. And this means the probability that 𝑍 is greater than negative three and less than one is 0.8413 minus 0.0013. That’s 0.84.

And to change from a decimal to a percentage, we multiply by 100. So 0.84 is the same as 84 percent. And we can say that the percentage of students that weigh between 54 and 70 kilograms is 84 percent.

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