Video: Estimating Normal Distribution Probabilities in Context

A crop of apples has a mean weight of 105 g and a standard deviation of 3 g. 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 99 g and 111 g?

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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 99 grams and 111 grams?

Remember, the graph of a curve representing the normal distribution with a mean of πœ‡ and a standard deviation 𝜎 is bell-shaped and symmetric about the mean, and the total area under the curve is 100 percent or one. It can be really useful to sketch the curve out to help you decide the best way to calculate probabilities.

The mean of the crop of apples here is 105 grams, and its standard deviation is three. The question is asking us to calculate the probability that a randomly selected apple has a weight between 99 grams and 111 grams. That’s represented by the area shaded. Once we sketch this out, we can calculate the 𝑍-value. This is a way of scaling our data or standardizing it, in what becomes a standard normal distribution. Once we complete this step, we can work from a single standard normal table.

Let’s look at the 𝑋 value of 111 grams. We can substitute 105 for πœ‡ and 𝜎 for three. And that gives us a 𝑍-value of 111 minus 105 all divided by three, which is two. We can therefore find the probability that a randomly selected apple has a weight of less than 111 grams by looking up a 𝑍-value of two in the table. The probability that 𝑍 is less than two is 0.9772. And in turn, the probability that 𝑋, the randomly selected apple, is less than 111 grams is also 0.9772.

Remember though, the question wanted to know the probability the weight was between 99 grams and 111 grams. We’re going to need to subtract the probability that the weight is less than 99 grams. Substituting 99 into our formula for the 𝑍-value and we get 99 minus 105 all divided by three, which is a 𝑍-value of negative two.

Since our standard normal table only contains positive values for 𝑍, we need to use the symmetry of the curve. And instead, we look up a 𝑍-value of two. Looking up a 𝑍-value of two though gives us the percentage between zero and two. So since the total area under the curve is 100 percent or one, we subtract the probability that 𝑍 is less than two from one to work out the bit that we’re interested in.

We already said that the probability that 𝑍 is less than two is equal to 0.9772. So the probability that 𝑍 is greater than two is one minus 0.9772, which is 0.0228. And in turn, that means that the probability that a randomly selected apple has a weight of less than 99 grams is 0.0228.

Remember, we wanted to know the probability that an apple had a weight between 99 and 111 grams. So in this case, we subtract the probability that 𝑋 is less than 99 from the probability that 𝑋 is less than 111. That’s 0.9772 minus 0.0228, which is 0.9544. Multiplying by 100 and then rounding to the nearest percent and we can see that the approximate probability that a randomly selected apple from the crop has a weight between 99 grams and 111 grams is 95 percent.

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