Find the z-score corresponding to a raw score of 132 from a normal distribution with mean 100 and standard deviation 15.

Duke nukem games. Solution

We compute

132 - 100
z = = 2.133
15

Example

A z-score of 1.7 was found from an observation coming from a normal distribution with mean 14 and standard deviation 3. Find the raw score.

Solution

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We have

x - 14
1.7 =
3

To solve this we just multiply both sides by the denominator 3,

(1.7)(3) = x - 14

5.1 = x - 14

x = 19.1

The z-score and Area

Often we want to find the probability that a z-score will be less than a given value, greater than a given value, or in between two values. To accomplish this, we use the table from the textbook and a few properties about the normal distribution.

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Example

Find

P(z < 2.37)

Solution

We use the table. Notice the picture on the table has shaded region corresponding to the area to the left (below) a z-score. This is exactly what we want. Below are a few lines of the table.

z.00.01.02.03.04.05.06.07.08.09
2.2.9861.9864.9868.9871.9875.9878.9881.9884.9887.9890
2.3.9893.9896.9898.9901.9904.9906.9909.9911.9913.9916
2.4.9918.9920.9922.9925.9927.9929.9931.9932.9934.9936

The columns corresponds to the ones and tenths digits of the z-score and the rows correspond to the hundredths digits. For our problem we want the row 2.3 (from 2.37) and the row .07 (from 2.37). The number in the table that matches this is.9911.

Hence

P(z < 2.37) = .9911

Example

Wallpaper wizard 2 1 52. Find

P(z > 1.82)

Solution

In this case, we want the area to the right of 1.82. This is not what is given in the table. We can use the identity

Record lectures 3 0. P(z > 1.82) = 1 - P(z < 1.82)

reading the table gives

P(z < 1.82) = .9656

Our answer is

P(z > 1.82) = 1 - .9656 = .0344

Example

Find

P(-1.18 < z < 2.1)

Solution

Once again, the table does not exactly handle this type of area. However, the area between -1.18 and 2.1 is equal to the area to the left of 2.1 minus the area to the left of -1.18. That is

P(-1.18 < z < 2.1) = P(z < 2.1) - P(z < -1.18)

To find P(z < 2.1) we rewrite it as P(z < 2.10) and use the table to get

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P(z < 2.10) = .9821.

The table also tells us that

10-3 Police Code

P(z < -1.18) = .1190

Now subtract to get

P(-1.18 < z < 2.1) = .9821 - .1190 = .8631

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