find and also interpret the area under a common curve uncover the value of a common random change

Finding locations Using a Table

Once we have the general idea the the common Distribution, the following step is to learn exactly how to find areas under the curve. We"ll learn two various ways - making use of a table and using technology.

You are watching: Find the area under the standard normal curve between z = 1 and z = 2.

Since every normally spread random variable has a slightly different distribution shape, the only way to find locations using a table is come standardize the variable - transform our change so it has a median of 0 and also a traditional deviation that 1. Just how do we execute that? Use the z-score!


As we noted in section 7.1, if the random variable X has a typical μ and also standard deviation σ, then transforming X making use of the z-score create a arbitrarily variable with typical 0 and standard deviation 1! v that in mind, we just need come learn how to find locations under the standard regular curve, which can then be applied to any normally dispersed random variable.

Finding Area under the conventional Normal Curve to the Left

Before we look a couple of examples, we need to first see exactly how the table works. Before we start the section, you require a copy the the table. You have the right to download a printable copy the this table, or usage the table in the ago of her textbook. It must look something prefer this:

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It"s pretty overwhelming at first, but if you look at the photo at the peak (take a minute and check that out), you deserve to see that it is indicating the area to the left. That"s the vital - the worths in the middle represent areas to the left the the equivalent z-value. To identify which z-value it"s introduce to, we look come the left to get the an initial two number and above to the columns to get the percentage percent value. (Z-values with more accuracy should be rounded to the hundredths in bespeak to use this table.)

Say we"re in search of the area left the -2.84. To perform that, we"d begin on the -2.8 row and go throughout until we gain to the 0.04 column. (See picture.)

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From the picture, we deserve to see the the area left the -2.84 is 0.0023.

Finding locations Using StatCrunch

Click on Stat > Calculators > Normal

Enter the mean, traditional deviation, x, and the direction that the inequality. Then push Compute. The image listed below shows P(Z

Example 1


a. Discover the area left the Z = -0.72

< expose answer >


The area left that -0.72 is approximately 0.2358.


b. Discover the area left of Z = 1.90

< expose answer >


The area left that 1.90 is approximately 0.9713.


Finding Area under the traditional Normal Curve to the Right

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To find locations to the right, we should remember the match rule. Take a minute and also look earlier at the rule from ar 5.2.

Since we recognize the entire area is 1,

(Area come the ideal of z0) = 1 - (Area come the left of z0)

Example 2


a. Uncover the area to the appropriate of Z = -0.72

< expose answer >


area best of -0.72 = 1 - (the area left of -0.72)
= 1 - 0.2358
= 0.7642

b. Find the area to the right of Z = 2.68

< disclose answer >


area right of 2.68 = 1 - (the area left that 2.68)
= 1 - 0.9963
= 0.0037

An alternate idea is to usage the symmetric residential or commercial property of the normal curve. Rather of looking come the appropriate of Z=2.68 in instance 2 above, we might have looked in ~ the area left the -2.68. Since the curve is symmetric, those areas are the same.

Finding Area under the standard Normal Curve in between Two Values

To discover the area in between two values, us think of it in 2 pieces. Mean we want to discover the area between Z = -2.43 and Z = 1.81.

What we carry out instead, is uncover the area left that 1.81, and also then subtract the area left the -2.43. Prefer this:


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–
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=

So the area between -2.43 and 1.81 = 0.9649 - 0.0075 = 0.9574

Note: StatCrunch is may be to calculation the "between" probabilities, so girlfriend won"t have to perform the calculation above if you"re making use of StatCrunch.

Example 3


a. Find the area in between Z = 0.23 and Z = 1.64.

< disclose answer >


area between 0.23 and 1.64 = 0.9495 - 0.5910 = 0.3585


b. Uncover the area in between Z = -3.5 and Z = -3.0.

< reveal answer >


area between -3.5 and -3.0 = 0.0013 - 0.0002 = 0.0011


Finding areas Under a common Curve making use of the Table

draw a map out of the typical curve and also shade the desired area. Calculation the matching Z-scores. Discover the matching area under the standard normal curve.

If friend remember, this is specifically what we saw happening in the Area the a Normal circulation demonstration. Follow the link and also explore again the relationship between the area under the conventional normal curve and also a non-standard common curve.

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Finding locations Under a typical Curve utilizing StatCrunch

Even despite there"s no "standard" in the location here, the directions space actually specifically the same as those indigenous above!

Click on Stat > Calculators > Normal

Enter the mean, conventional deviation, x, and also the direction of the inequality. Then press Compute. The image listed below shows P(Z What proportion of people are geniuses? Is a systolic blood pressure of 110 unusual? What percent of a certain brand that light pear emits between 300 and also 400 lumens? What is the 90th percentile for the weights of 1-year-old boys?

All of these questions have the right to be answered using the normal distribution!

Example 4


Let"s take into consideration again the distribution of IQs that we looked at in instance 1 in ar 7.1.

We observed in that example that tests because that an individual"s knowledge quotient (IQ) space designed come be typically distributed, with a typical of 100 and a typical deviation that 15.

We also saw that in 1916, psychologist Lewis M. Thurman set a tip of 140 (scaled come 136 in today"s tests) because that "potential genius".

Using this information, what percentage of individuals are "potential geniuses"?

Solution:

attract a map out of the common curve and also shade the wanted area.
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calculation the matching Z-scores.
Z = X - μ = 136 - 100 = 2.4
σ 15
find the equivalent area under the typical normal curve. P(Z>2.4) = P(Z

Based on this, it looks like around 0.82% that individuals can be defined as "potential geniuses" according to Dr. Thurman"s criteria.


Example 5


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Source: stock.xchng

In example 2 in ar 7.1, we were told that weights of 1-year-old guys are around normally distributed, with a average of 22.8 lbs and a traditional deviation of around 2.15. (Source: About.com)

If we randomly select a 1-year-old boy, what is the probability that he"ll sweet at the very least 20 pounds?

Solution:

Let"s execute this one using technology. We should still begin with a sketch:

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Using StatCrunch, we obtain the adhering to result:

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According to these results, the looks choose there"s a probability of around 0.9036 that a randomly selected 1-year-old boy will certainly weigh much more than 20 lbs.


Why don"t you try a couple?

Example 6


Photo: A Syed

Suppose the the volume of repaint in the 1-gallon repaint cans produced by Acme Paint agency is about normally spread with a typical of 1.04 gallons and a traditional deviation that 0.023 gallons.

What is the probability the a randomly selected 1-gallon deserve to will in reality contain at least 1 gallon the paint?

< reveal answer >


In this case, we desire P(X ≥ 1). Using StatCrunch again, we obtain the complying with result:

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According come the calculation, it looks like the probability the a randomly selected have the right to will have an ext than 1 gallon is approximately 0.9590.


Example 7


Suppose the amount of irradiate (in lumens) emitted by a certain brand the 40W light bulbs is normally spread with a typical of 450 lumens and also a conventional deviation of 20 lumens.

What percent of bulbs emit in between 425 and 475 lumens?

< expose answer >


To answer this question, we should know: P(425

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P(X

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So P(425 What is the 90th percentile because that the weights the 1-year-old boys? What IQ score is below 80% of every IQ scores? What weight does a 1-year-old boy must be therefore all however 5% the 1-year-old boys weight less than he does?

As v the previous varieties of problems, we"ll learn exactly how to perform this utilizing both the table and technology. Make certain you know both methods - they"re both supplied in many fields the study!

Finding Z-Scores using the Table

The idea here is that the values in the table represent area come the left, therefore if we"re inquiry to find the value with an area that 0.02 to the left, us look for 0.02 on the inside that the table and find the matching Z-score.

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Since us don"t have an area that exactly 0.02, we need to think a bit. We have two choices: (1) take the the next area, or (2) average the two values if it"s equidistant indigenous the 2 areas.

In this case, it"s nearly equidistant, so we"ll take the average and say that the Z-score matching to this area is the mean of -2.05 and also -2.06, for this reason -2.055.

Finding Z-Scores making use of StatCrunch

Click on Stat > Calculators > Normal

Enter the mean, typical deviation, the direction that the inequality, and the probability (leave X blank). Then press Compute. The image below shows the Z-score with an area that 0.05 come the right.

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Let"s try a few!

Example 8


Using the normal calculator in StatCrunch, we gain the adhering to result:

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So the Z-score with an area the 0.90 come the left is 1.28. (We generally round Z-scores come the hundredths.)


b. Find the Z-score through an area the 0.10 to the right.

< disclose answer >


This is actually the very same value as instance 7 above! one area of 0.10 come the right means that that must have an area that 0.90 to the left, so the price is again 1.28.


c. Discover the Z-score such the P( Z 0 ) = 0.025.

< expose answer >


Using StatCrunch, we gain the complying with result:

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So the Z-score is -1.96.



So we"ve talked around how to find a z-score given an area. If friend remember, the an innovation instructions didn"t specify that the distribution needed to be the standard typical - we actually uncover values in any normal circulation that correspond to a given area/probability making use of those same techniques.

Example 9


Referring come IQ scores again, with a average of 100 and a standard deviation that 15. Uncover the 90th percentile for IQ scores.

Solution:

First, we need to analyze the difficulty into one area or probability. In section 3.4, we said the kth percentile of a collection of data divides the reduced k% that a data set from the upper (100-k)%. Therefore the 90th percentile divides the lower 90% from the upper 10% - definition it has about 90% listed below and about 10% above.

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Using StatCrunch, we acquire the following result:

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Therefore, the 90th percentile because that IQ scores is around 119.


Example 10



Photo: A Syed

This would be the worth with just 5% much less than it. Using StatCrunch, we have actually the following result:

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Based top top this calculation, the Acme Paint firm can say the 95% the its can be ~ contain at the very least 1.002 gallons the paint.



Example 11


Using StatCrunch again, we discover the value v an area of 0.95 come the left:

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So a 1-year-old boy would need to weigh about 26.3 lbs. Because that all however 5% of all 1-year-old boys to weigh less than he does.



Finding zα

The notation zα ("z-alpha") is the Z-score with an area the α to the right.

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The principle of zαis used generally throughout the remainder the the course, for this reason it"s an essential one to it is in comfortable with. The applications won"t be automatically obvious, but the significance is that we"ll it is in looking for occasions that room unlikely - and so have actually a very small probability in the "tail".