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 less than 105 grams?
In a normal distribution, there’s this bell-shaped curve, and the mean is 105 and the standard deviation is three. So the deviation means it would deviate from the 105 by three, so maybe three above 105 at 108 or three below at 102.
You could go another three further to the right to be at 111 — that would be two standard deviations away — and same on the left. It could be two standard deviations away and be at 99. And we can also be three standard deviations away at 114 or at 94.
So the usefulness is knowing the percentages that fall within these deviations, the standard deviations. Approximately 60% of the data fall within one stan- standard deviation of the mean. So splitting that in half, 34 percent would be a standard deviation above and then 34 percent would be below, and then approximately 95 percent of the data, all of the data, fall within two standard deviations of the mean.
So if we take 95 and subtract 68, that means there’s — so if we take 95 and then subtract 68, that means there’s 27 percent to split between these two columns, and if we split that equally in half, then 13.5 percent will be in each of these columns.
And now 99.7 percent of all of the data fall within three standard deviations of the mean. So if we take 99.7 subtract 95, we get 4.7. And then splitting that in half, each of these columns will contain 2.35 percent of all of the data.
And then including the last two pieces, that would be 100 percent of the data. So if we take 100 minus 99.7, we have 0.3. And dividing that by two means the rest of each of these sides will contain 0.15 percent of all of the data.
So the question asks what is the approximate probability that a randomly selected apple from the crop has a weight less than 105 grams, which would be everything to the left. So the probability of being less than 105 grams, we would add the 0.15 percent, the 2.35 percent, the 13.5 percent, and the 34 percent to get 50 percent.
So did we have to go through all of these steps to know that half of it would be below the average, the mean? Probably not, because we knew it’s the normal, but it’s always good to just double-check, going very specifically through every single step.
So again, the approximate probability that a randomly selected apple from this crop has a weight that’s less than 105 grams is 50 percent.