Video: Estimating Normal Distribution Probabilities in Context

A crop of apples has a mean weight of 105g and a standard deviation of 3g. 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 102g and 108g?

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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 πœ‡ 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 β€” that’s the measure of spurt here β€” 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, πœ‡, 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 β€” we have a mean of zero β€” 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.

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