Finding Areas Using a Table
Once we have the general idea of the Typical Distribution, the following step is to learn just how to discover areas under the curve. We"ll learn two different methods  making use of a table and making use of technology.
You are watching: Find the area under the standard normal curve to the left of zequals1.25.
Due to the fact that eexceptionally typically distributed random variable has actually a slightly various circulation shape, the only way to find areas utilizing a table is to standardize the variable  transcreate our variable so it has actually a expect of 0 and also a conventional deviation of 1. How do we carry out that? Use the zscore!
As we noted in Section 7.1, if the random variable X has a expect μ and also conventional deviation σ, then transforming X using the zscore creates a random variable via mean 0 and typical deviation 1! With that in mind, we simply have to learn how to find areas under the standard normal curve, which deserve to then be applied to any usually distributed random variable.
Finding Area under the Standard Regular Curve to the Left
Before we look a couple of examples, we should first view exactly how the table functions. Before we start the area, you require a copy of the table. You deserve to downpack a printable copy of this table, or usage the table in the earlier of your textbook. It should look something favor this:
It"s pretty overwhelming at initially, but if you look at the photo at the peak (take a minute and check it out), you deserve to see that it is indicating the area to the left. That"s the essential  the worths in the middle represent areas to the left of the corresponding zvalue. To identify which zworth it"s referring to, we look to the left to gain the initially 2 digits and over to the columns to acquire the hundredths worth. (Zvalues through more accuracy need to be rounded to the hundredths in order to use this table.)
Say we"re looking for the area left of 2.84. To execute that, we"d begin on the 2.8 row and go throughout until we get to the 0.04 column. (See image.)
From the photo, we deserve to check out that the area left of 2.84 is 0.0023.
Finding Areas Using StatCrunch
Click on Stat > Calculators > Normal Enter the expect, standard deviation, x, and also the direction of the inequality. Then press Compute. The photo listed below reflects P(Z Example 1 a. Find the location left of Z = 0.72 < expose answer > The area left of 0.72 is around 0.2358. b. Find the location left of Z = 1.90 < expose answer > The location left of 1.90 is roughly 0.9713. Finding Area under the Standard Common Curve to the RightTo uncover locations to the best, we need to remember the enhance rule. Take a minute and look ago at the dominance from Section 5.2. Because we know the whole area is 1, (Area to the right of z0) = 1  (Area to the left of z0) Example 2 a. Find the area to the right of Z = 0.72 < disclose answer >
b. Find the location to the appropriate of Z = 2.68 < disclose answer >
An alternate idea is to use the symmetric home of the normal curve. Instead of looking to the best of Z=2.68 in Example 2 over, we could have actually looked at the location left of 2.68. Due to the fact that the curve is symmetric, those areas are the exact same. Finding Area under the Standard Normal Curve Between Two ValuesTo discover the area between 2 worths, we think of it in 2 pieces. Suppose we want to uncover the location between Z = 2.43 and also Z = 1.81. What we execute instead, is find the area left of 1.81, and also then subtract the location left of 2.43. Like this:
So the area between 2.43 and 1.81 = 0.9649  0.0075 = 0.9574 Note: StatCrunch is able to calculate the "between" probabilities, so you will not should percreate the calculation above if you"re using StatCrunch. Example 3 a. Find the location between Z = 0.23 and also Z = 1.64. < reveal answer > area in between 0.23 and also 1.64 = 0.9495  0.5910 = 0.3585 b. Find the area in between Z = 3.5 and also Z = 3.0. < disclose answer > location in between 3.5 and 3.0 = 0.0013  0.0002 = 0.0011 Finding Areas Under a Common Curve Using the TableDraw a sketch of the normal curve and also shade the wanted location. Calculate the matching Zscores. Find the matching area under the standard normal curve.If you remember, this is precisely what we witnessed happening in the Area of a Typical Distribution demonstration. Follow the attach and check out aobtain the partnership in between the area under the standard normal curve and also a nontypical normal curve. Finding Areas Under a Typical Curve Using StatCrunchEven though there"s no "standard" in the title here, the directions are actually precisely the exact same as those from above!
