Find the standard normal distribution curve to the left of z= -1.26

Question

Find the standard normal distribution curve to the left of z= -1.26

Answer 1

Hope, this is a great question to test your ability to understand exactly how Z-scores relate to a normal distribution curve, and you can almost estimate it without any work (a good way to check your answer). Here are key facts from your question:

1. You have the standard normal distribution, commonly noted as N(μ,σ) = N(0,1) where μ is the mean and σ is the standard deviation. This function forms the "classic" picture of a normal curve. You have a mean of 0, and a standard deviation of 1... this makes things a bit easier.
2. Remember, the area under the curve for a normal distribution must be equal to 1.
3. Your z-score is negative so you are looking at a value that is below the mean.
4. Z-scores are critical in the basic understanding of a normal curve. +1 σ from your mean would relate to a Z-score of + 1. So, you can think of this question asking you what the probability of finding an x-value in the region to the left of z = -1.26.
5. If you center the normal distribution at μ=0, you immediately know that at least 50% of your area under the curve is "gone", so your value must be less than .500
6. If you move 1 standard deviation to the left of the mean you can be sure that you will lose another ~34% of the area under the curve. Recall, 68% of the area under the curve is between + 1σ, so half of that would be on the left of the mean, in addition to the other 50% we just mentioned. You know your probably must be less than 1 - (50% + 34%) = 16%.
7. Since your Z-score is greater than z = -2, you know that no more than 95% of the area under the curve can be gone. If you read that and thought... what? Here is what you should remember about your standard normal curve. About 68% of all area under the curve is between + 1 standard deviation of the mean, split evenly on both sides (about 34% in the "negative" and "positive"). In addition, About 95% of all area under the curve is between + 2 standard deviations of the mean. We know that since the z-score here is -1.26, you will for sure have lost all of the "positive" side of the mean, and then at least 1 standard deviation of "negative side", which we described in step 6. With our z-score being within 2 standard deviations, we know we haven't hit the "95%" mark yet, so you can confidently say that you still have at least 5% of the area under the curve available. Your value must be greater than 5%

So, without doing any real math you know your P(Z<z=-1.26) ∈ (5%, 16%). This is good for checking your answer.

Now you can simply use the Left-Sided Z-Table to look up your answer. Your statistics textbook will have one inside of it, or you can simply Google it. Remember, the standard format for a Z-table is to have the 0.1 level on the rows, and 0.01 level in the columns. Look for where row Z=-1.2 intersects the column for 0.06. This will be your answer, or roughly P(Z< z=-1.26) = 0.104 (10.4%)... Which falls into the (5%, 16%) range we referenced above.

Answer 2