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Solving Problems Involving Normal Curves Review Sheet

Normal Curve Characteristics:

Any Normal Z (Stnd Normal) X ~ N(,)

Shape Symmetric mound Symmetric mound

Center Mean of Mean of 0

Spread Std Dev of Std Dev of 1

Mean = Median = Mode (middle of the graph)

Area to left or right of mean is 50%; total area under the curve adds to 1

Z-Scores:

Positive values are above the mean and negative values are below

Formula: z = -

x -

------- = number of standard deviations (), x is away from mean

When comparing separate events, the smaller of two z scores is worse

Example: If Jon scores a 92 on a test with a mean of 83 and a standard deviation of 6,

what is his z-score.

a) z = -

x - 92 - 83 9

------- = ------------ = ---- = 1.5

6 6

Z-Table:

Measures the area to the left of a value. For example, z = 1.68

gives us a value of 0.9535, which mean 95.35% of the area under the curve is to the left of

1.68 (smaller than it)

Distance Within Area

Empirical Rule: also known as 68-95-99.7 Rule 1 Stnd Dev 68%

A normal curve will have the following percentages

of its area within set distance from the mean 2 2 Stnd Dev 95%

3 3 Stnd Dev 99.7%

Example:

If 68% of the scores on the SOL lie between 388 and 432, what is the mean and standard

deviation of the SOL scores.

b) 432 - 388 = 44 = 2 ; so the standard deviation, = 22.

The mean, , lies halfway between 432 and 388 or 410.

Find probabilities (area under the curve): 2nd Vars 3: normalcdf(LB,UB, ,)

2nd VARS

a ab b

Using P(x < a) P(a < x < b) P( x > b)

Z-table Za value from table Zb - Za value from table 1 - Zb value from table

Calculator Normcdf(-E99,a,,) Normcdf(a,b,,) Normcdf(b,E99,,)

Remember that the mean and standard deviation of a Z distribution is (0,1). Draw the

curve and shade in the area that you are looking for. This will help determine which

bound (upper or lower) that we have in the problem. If we only have one bound, then if

we have an upper bound (figure on the left) we use -E99 as the lower bound. If we have

a lower bound (figure on the right), then we use E99 as the upper bound. We get to the

E# by using the 2nd "," (comma) key on our calculator.

Word problems finding the (normal) probability

In most word problems the mean and standard deviation are clearly given to us in the

problem. We need to figure out which bounds we are given. Pay attention to the words

less than (< - picture on the left above) and more than (> - picture on the right above). If

we are given two bounds, then the smallest is the lower and the largest is the upper.

Example:

In a gym class students have to run a mile. For a 6th grade class the average was 512

seconds with a standard deviation of 68.

c) What is the probability of a student running less than 400 seconds? 400

normalcdf(-E99,400,512,68) = 0.0498 512 610

d) What is the probability of a student running more than 610 seconds?

normalcdf(610,E99,512,68) = 0.0748 512

e) What is the probability of a student running between 475 and 525 seconds?

normalcdf(475,525,512,68) = 0.2826 475 525

Given the area (percentile) and find a number corresponding

512

In some word problems they give us the mean and standard deviation and the ask what

value corresponds to a certain percentage or a percentile. These problems are the

inverse of the probability problems and we use the invnorm function (option 3 from 2nd

Vars) of the calculator and the calculator will give us the value corresponding to that

percentile. Invnorm (percentile (in decimal form), mean, standard deviation)

Example:

f) On a math test which had a mean of 83 and a standard deviation of 6, what is the

90th percentile score? Invnorm(0.90,83,6) = 90.69

g) On a math test which had a mean of 83 and a standard deviation of 6, what is the

45th percentile score? Invnorm(0.45,83,6) = 82.25

Worksheet Problems:

1. (Ref a) Find the following z-scores with a mean of 25 and a standard

deviation of 4:

a) 20 b) 32 c) 28 d) 25

2. (Ref a) If Sarah scored 78 on her History test which had a mean of 70 and

a standard deviation of 3 and she scored 84 on her Math test which had

a mean of 80 and a standard deviation of 2, on which test did she score

better?

3. (Ref a) Find the following z-scores with a mean of 20 and a standard

deviation of 5:

a) 20 b) 32 c) 28 d) 25

4. (Ref a) Ted Williams was the last player to hit over .400 (.406 in 1941,

mean .26648 and standard deviation of 0.051). Since then George Brett

has come the closest, hitting .390 in 1980, mean average of .26907 and a

standard deviation of 0.036. Who had a better year?

5. (Ref b) If 2.5% of scores on a normally distributed college entrance test

were below 60% and 2.5% of the scores were above 84%, what was the

mean and the standard deviation of the test? (Hint: empirical rule)

How do you calculate the area under a curve? Decide how many pieces you want to break the curve into. How about 100? ... Set the area to zero (chickensÂ²). ... Start with the initial x-value (in the example Iâ€™ve been using â€” thatâ€™s x = 1). Calculate the height of the rectangle. ... Find the area of this rectangle and add it to the total area. Move on the next x-value and repeat until you get to the final x.

Title: Reflections

Author: Chris Headlee

Creator: Microsoft® PowerPoint® 2010

Producer: Microsoft® PowerPoint® 2010

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