Video: Using Tables to Calculate Probabilities of the Normal Distribution

Use tables to find the normal probability corresponding to a 𝑧-score of −1.73.

02:37

Video Transcript

Use tables to find the normal probability corresponding to a 𝑧-score of negative 1.73.

Now, at first glance, this might appear fairly straightforward. But notice how our 𝑧-score is negative. So we’re going to need to be a little bit clever about how we read the values from our standard normal table. Remember, normally distributed data can be represented using this bell curve. It’s completely symmetrical about the mean. And the area underneath the curve is one or 100 percent.

When we find the 𝑧-score of our random variable, we’re essentially standardising our data. And it allows us to read any values off of the standard normal table. And when we do so, we’re looking at the normal distribution with a mean of zero and a standard deviation of one.

So negative 1.73 is going to be on the left-hand side of our bell curve. And of course, our standard normal tables are cumulative. We’re going to be finding the probability that 𝑧 is less than negative 1.73. And then, we need to consider the symmetry of the curve because our standard normal tables don’t include negative values for 𝑧. So we see that the probability that 𝑧 is less than negative 1.73 must be equal to the probability that 𝑧 is greater than 1.73.

However, if we look up a 𝑧-score of 1.73 in our standard normal table, it will tell us the probability that 𝑧 is less than 1.73. It’s everything to the left of that line. We said earlier though that the total area under the curve is one. So we can find this area by subtracting the probability that 𝑧 is less than 1.73 from one. And let’s look at snapshots of the standard normal table so we can work out how to read the probability that 𝑧 is less than 1.73 from this table.

The first two digits of the number we’re interested in are 1.7. So we’re interested in all the numbers on this row. And the third digit is a three. So we’re actually interested in the digits in this column. We can see that the value in this row and this column is 0.9582. So the probability that 𝑧 is less than 1.73 is 0.9582. And to find the probability that 𝑧 is less than negative 1.73, we’re going to subtract this from one. And when we do, we get 0.0418.

And the probability that 𝑧 is less than negative 1.73 or the probability corresponding to a 𝑧-score of negative 1.73 is 0.0418.

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