Using the Z transformation to find probabilities

- In a previous section, we discussed the Z-score. We were able to figure out how far from the mean, our data point was, in terms of standard deviations. To recap, if I gave you a mean of 150, and a standard deviation of 25. And then I gave you a data point of 211, you could find the Z-score of 2.44. On our normal curve, that would be over here, far to the right of the mean. Now let's take this one step further.

If we assume that the data is normally distributed, what percentage of men, weigh more then 211 pounds. Well, if we use the standard normal distribution table, we can find our Z of 2.44. Go down on the left, and find 2.4. Then, move across to the column labeled 0.04. That intersection is the value of 2.44. At that intersection, we find the number 0.9927, What does that mean? Well, it tells us that according to our mean and standard deviation, 99.27% of all men weigh 211 pounds or less, which means that the probability a man weighs more then 211 pounds is only 0.73%.

Let's use the Z transformation to answer another question. What is the probability a man weighs between 140 pounds, and 170 pound. Now we need two Z-scores. First, the Z-score for 170 pounds, plugging our numbers into the formula, we get a Z-score of 0.80. The chart tells us the value for this is 0.7881, or 78.81% Now, the Z-score for 140 pounds.

Plugging our numbers into this formula, here we get a Z-score of negative 0.40. How do we find that on the chart? Well, for this negative value, you find 0.40. The chart tells us the value for this is 0.6554. Because the Bell Curve is symmetrical, we can subtract this number from 1.0, to find where we are on the left-hand side of the mean.

So, one minus 0.6554, gives us 0.3446, or 34.46%. Remember, we are trying to find the probability a man weighs between 140 pounds and 170 pounds. Well, the probability a man weighs 170 pounds or less, is 78.81%.

The probability a man weighs 140 pounds or less, is 34.46%. So, by subtracting those two percentages, we now know that the probability that a man weighs between 140 pounds and 170 pounds, is 44.35%. Means, standard deviations, normal curves, Z-scores, and Z-score tables. Not only do you know what they mean, you're now able to calculate interesting probabilities all on your own.