The lengths of cylinders produced at a factory follow a normal distribution with mean 72 centimeters and standard deviation five centimeters. A cylinder is acceptable for sale if its length is between 64.4 centimeters and 73.4 centimeters. If a random sample of 1000 cylinders is chosen, how many cylinders would be acceptable for sale?
Our dataset is normally distributed. It has a mean 𝜇 of 72 centimeters and a standard deviation 𝜎 of five centimeters. In order to decide how many cylinders satisfy the criteria, we need to first work out the percentage of cylinders that will measure between 64.4 centimeters and 73.4 centimeters. To do this, we consider the probability that a randomly chosen cylinder is greater than 64.4 and less than 73.4. And the variable that we use to describe this is 𝑥.
So how do we find this probability? Well to begin with, we need to find the associated 𝑧 scores for 64.4 and 73.4. We use this formula 𝑥 minus 𝜇 over 𝜎. And what this does is essentially scale or standardize the data. And it means we can calculate probabilities using the standard normal table. And this standard normal curve has a mean 𝜇 of zero and a standard deviation 𝜎 of one. For an 𝑥-value of 64.4, the corresponding 𝑧-value is found by substituting 64.4 into the formula. It’s 64.4 minus 72 all over five. That’s negative 1.52.
And for an 𝑥-value of 73, our formula becomes 73.4 minus 72 again over five. And that’s 0.28. So in fact, we need to find the probability that 𝑧 is greater than negative 1.52 and less than 0.28. And let’s consider the shape of the standard normal curve to help us. It’s bell-shaped. And it’s completely symmetrical about the mean. And for the standard normal curve, we said that the mean is zero. The total area under the curve is one. And in fact, we’re interested in the area represented by the region between negative 1.52 and 0.28. That’s the one shaded.
Now, the probability is actually cumulative. So if we look at the 𝑧-value of 0.28 in our standard normal table, it will tell us the probability that 𝑧 is less than 0.28. That’s everything to the left of 0.28 on our curve. To find the probability represented by the shaded region then, we’re going to need to subtract the probability that 𝑧 is less than negative 1.52. And finding the probability that 𝑧 is less than 0.28 from the standard normal table is fairly straightforward, it’s 0.6103. But finding the probability that 𝑧 is less than negative 1.52 is a little more complicated. And that’s because our standard normal table has no negative values.
So at this point, we need to consider the symmetry of our normal curve. Now remember, this curve is completely symmetrical about the mean. So this means that the probability that 𝑧 is less than negative 1.52 is the same as the probability that 𝑧 is greater than 1.52. And since the probabilities are cumulative and the area under the curve is one, we can find this probability by subtracting the probability that 𝑧 is less than 1.52 from one.
And once we know this, we can find the probability that 𝑧 is less than 1.52 by finding 1.52 on our standard normal table. That gives us 0.9357. So the probability we’re interested in is found by subtracting one minus 0.9357 from 0.6103. And if we perform that calculation, we get 0.546. So how does this help us? Well, we have found the probability that 𝑧 is greater than negative 1.52 and less than 0.28. But in fact, we said that that is the same as the probability that 𝑥 is greater than 64.4 and less than 73.4. It’s the probability that a randomly chosen cylinder has a length between 64.4 centimeters and 73.4 centimeters.
We are choosing a random sample of 1000 cylinders. So how many will we expect that satisfy this criteria? Well, that’s found by multiplying 1000 by that probability by 0.546. And 1000 multiplied by 0.546 is 546. And we see that, out of a random sample of 1000 cylinders, 546 would satisfy the criteria and, therefore, would be acceptable for sale.