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Use tables to find the normal probability corresponding to a 𝑧-score of 2.13.

We are asked to find the normal probability corresponding to a 𝑧-score of 2.13, which means the proportion of points or the area that lies to the left of this value of 2.13 under the standard normal distribution curve. So here are our statistical tables for the standard normal distribution. Now, these tables give the proportion of points or the area that lies between zero and a positive 𝑧-score. That’s only the part of the area now shaded in pink on our figure. That’s okay though because we know that the normal distribution is completely symmetrical about its mean. And so the orange part of the area is exactly 0.5. We therefore need to add 0.5 to whatever value we find in our table.

Now, looking at our tables, we can see that they have 𝑧-scores ranging from zero to three in the first column. These values increase by 0.1 each time. And then in the top row of the table, we have options for the second decimal place of our 𝑧-score. The 𝑧-score we want to look up is 2.13, so we look up 2.1 in the first column and then 0.03 because 2.1 plus 0.03 gives 2.13. We then find the value in the cell of the table where this row and this column intersect, and it is 0.4834. This tells us that the area between zero and 2.13 is 0.4834. The total area to the left of 2.13 is 0.5 plus this value, which is 0.9834. This is the normal probability corresponding to a 𝑧-score of 2.13. And it represents the total area to the left of 2.13 under the standard normal curve.

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