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 this crop has a weight greater than 111 grams?
We’ve seen those keywords normal distribution. And so we draw a normal bell-shaped curve to help represent the distribution of the weights. The normal distribution curve is centered on the mean of the normal distribution 𝜇, which we’re told in the question is 105 grams. The natural unit along the bottom is 𝜎, the standard deviation, which we’re told is three grams.
So, for example, one standard deviation above the mean, which is 105, is 108. And we can fill in the other values marked too. We see the values 99, 102, and 111 joining 105 and 108. We’re looking for the approximate probability that a randomly selected apple from this crop has a weight greater than 111 grams.
Luckily, we already have 111 marked. And so we can see that the probability that the apple’s weight is greater than 111 grams is represented by this region that I’ve shaded. It’s a general fact about normal distributions that 34 percent of the data lie between the mean and one standard deviation above the mean. It’s the same proportion one standard deviation below the mean to the mean. This is due to the symmetry of the bell curve.
There’s another 13.5 percent of the data in this region from one standard deviation above the mean to two standard deviations above the mean. And by symmetry, the region from two standard deviations below the mean to one standard deviation below the mean also has this proportion of the data. The rest of the data must lie in the orange and pinky-purple shaded regions.
How much of the data is that? Well, it’s all the data minus that which has already been accounted for. We have to subtract two lots of 34 percent and two lots of 13.5 percent. And that leaves us with five percent of the data shared between the two shaded regions.
And by symmetry, the two shaded regions must have the same proportion of data. And so the proportion of points in the orange shaded region which represent the probability that a randomly selected apple from the crop has weight greater than 111 grams is 2.5 percent.
Actually, this is just an approximate value as all the percentages that we’ve been working with have been rounded. This value is very close to the value of 2.35 percent, which is the proportion of the points that lie between two standard deviations and three standard deviations above the mean.
In the region above the underline, there are 2.35 percent of the data. But we also need to include the 0.15 percent of data, which lies in the long tail more than three standard deviations above the mean.