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Normal Probability Distribution

The Normal Distribution functions:

#1: normalpdf pdf = Probability Density Function

This function returns the probability of a single value of the random variable x. Use this to graph a

normal curve. Using this function returns the y-coordinates of the normal curve.

Syntax: normalpdf (x, mean, standard deviation)

#2: normalcdf cdf = Cumulative Distribution Function

This function returns the cumulative probability from zero up to some input value of the random

variable x. Technically, it returns the percentage of area under a continuous distribution curve from

negative infinity to the x. You can, however, set the lower bound.

Syntax: normalcdf (lower bound, upper bound, mean, standard deviation)

#3: invNorm( inv = Inverse Normal Probability Distribution Function

This function returns the x-value given the probability region to the left of the x-value.

(0 < area < 1 must be true.) The inverse normal probability distribution function will find the

precise value at a given percent based upon the mean and standard deviation.

Syntax: invNorm (probability, mean, standard deviation)

Example 1: Given a normal distribution of values for which the mean is 70 and the standard deviation is 4.5.

Find:

a) the probability that a value is between 65 and 80, inclusive.

b) the probability that a value is greater than or equal to 75.

c) the probability that a value is less than 62.

d) the 90th percentile for this distribution.

(answers will be rounded to the nearest thousandth)

1a: Find the probability that a value is between 65 and 80,

inclusive. (This is accomplished by finding the probability of the

cumulative interval from 65 to 80.)

Syntax:normalcdf(lower bound, upper bound, mean, standard deviation)

Answer: The probability is

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1b: Find the probability that a value is greater than or equal to 75. (The upper boundary in this problem will be positive infinity. The largest

value the calculator can handle is 1 x 1099. Type 1 EE 99. Enter the EE by

pressing 2nd, comma -- only one E will show on the screen.)

Answer: The probability is

1c: Find the probability that a value is less than 62. (The lower boundary in this problem will be negative infinity. The smallest

value the calculator can handle is -1 x 1099. Type -1 EE 99. Enter the EE by

pressing 2nd, comma -- only one E will show on the screen.)

Answer: The probability is

1d: Find the 90th percentile for this distribution. (Given a probability region to the left of a value (i.e., a percentile), determine

the value using invNorm.)

Answer: The x-value is

Example 3: Graph and examine a situation where the mean score is 46 and the standard

deviation is 8.5 for a normally distributed set of data.

Go to Y= .

Adjust the window.

GRAPH.

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Practice and Homework (Independent # odd and HW even #)

1. The amount of mustard dispensed from a machine at The

Hotdog Emporium is normally distributedwith a mean of

0.9 ounce and a standard deviation of 0.1 ounce. If the

machine is used 500 times, approximately how many times

will it be expected to dispense 1 or more ounces of

mustard.

Choose:

5 16 80 100

Answer

2. Professor Halen has 184 students in his college

mathematics lecture class. The scores on the midterm

exam are normally distributed with a mean of 72.3

and a standard deviation of 8.9. How many students

in the class can be expected to receive a score

between 82 and 90? Express answer to the nearest

student.

Answer

3. A machine is used to fill soda bottles. The amount of soda

dispensed into each bottle varies slightly. Suppose the

amount of soda dispensed into the bottles is normally

distributed. If at least 99% of the bottles must have

between 585 and 595 milliliters of soda, find the greatest

standard deviation, to the nearest hundredth, that can be

allowed.

Answer

4. Residents of upstate New York are accustomed to large

amounts of snow with snowfalls often exceeding 6

inches in one day. In one city, such snowfalls were

recorded for two seasons and are as follows (in inches):

8.6, 9.5, 14.1, 11.5, 7.0, 8.4, 9.0, 6.7, 21.5, 7.7, 6.8, 6.1,

8.5, 14.4, 6.1, 8.0, 9.2, 7.1

Answer

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What are the mean and the population standard deviation

for this data, to the nearest hundredth?

5. Neesha's scores in Chemistry this semester were rather

inconsistent: 100, 85, 55, 95, 75, 100.

For this population, how many scores are within one standard

deviation of the mean?

Answer

6. Battery lifetime is normally distributed for large samples.

The mean lifetime is 500 days and the standard deviation is

61 days. To the nearest percent, what percent of batteries

have lifetimes longer than 561 days?

Answer

7. The number of children of each of the first 41

United States presidents is given in the

accompanying table. For this population,

determine the mean and the standard

deviation to the nearest tenth.

How many of these presidents fall within one

standard deviation of the mean?

Answer

8. From 1984 to 1995, the winning scores for a golf

tournament were 276, 279, 279, 277, 278, 278, 280, 282,

285, 272, 279, and 278. Using the standard deviation for

Answer

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this sample, Sx, find the percent of these winning scores

that fall within one standard deviation of the mean.

9. A shoe manufacturer collected data regarding men's shoe

sizes and found that the distribution of sizes exactly fits

the normal curve. If the mean shoe size is 11 and the

standard deviation is 1.5, find:

a. the probability that a man's shoe size is greater than or

equal to 11.

b. the probability that a man's shoe size is greater than or

equal to 12.5.

c.

Answer

10. Five hundred values are normally distributed with a mean

of 125 and a standard deviation of 10.

a. What percent of the values lies in the interval 115 - 135,

to the nearest percent?

b. What percent of the values is in the interval 100 - 150,

to the nearest percent?

c. What interval about the mean includes 95% of the data?

d. What interval about the mean includes 50% of the data?

Answer

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11. A group of 625 students has a mean age of 15.8 years with a

standard deviation of 0.6 years. The ages are normally

distributed. How many students are younger than 16.2

years? Express answer to the nearest student?

Binomial Probability

"Exactly", "At Most", "At Least"

Example 4

Problem used for demonstration: A fair coin is tossed 100 times. What is the probability that:

a. heads will appear exactly 52 times?

b. there will be at most 52 heads?

c. there will be at least48 heads?

Dealing with "Exactly":

A fair coin is tossed 100 times. What is the probability that:

part a. heads will appear exactly 52 times?

We have seen that the formula used with Bernoulli trials (binomial probability) computes

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the probability of obtaining exactly "r" events in "n" trials: where n = number of trials, r = number of specific events you wish to obtain, p = probability that the

event will occur, and q = probability that the event will not occur (q = 1 - p, the complement of the

event).

We have also seen that the built-in command binompdf(binomial probability density

function) can also be used to quickly determine "exactly".

(Remember, the function binompdf is found under

DISTR (2nd VARS), arrow down to #0 binompdf

and the parameters are:

binompdf (number of trials, probability of occurrence, number of specific events)

binompdf (n, p, r)

Here is our answer to part a.

Dealing with"At Most":

A fair coin is tossed 100 times. What is the probability that:

part b. there will be at most 52 heads?

The formula needed for answering part b is :

There is a built-in command binomcdf(binomial cumulative density function) that can be

used to quickly determine "at most". Because this is a "cumulative" function, it will find

the sum of all of the probabilities up to, and including, the given value of 52.

(The function binomcdf is found under

DISTR (2nd VARS), arrow down to #A binomcdf

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and the parameters are:

binomcdf (number of trials, probability of occurrence, number of specific events)

binomcdf (n, p, r)

Here is our answer to part b.

Dealing with "At Least":

A fair coin is tossed 100 times. What is the probability that:

part c. there will be at least48 heads?

The formula needed for answering part b is :

Keep in mind that "at least" 48 is the complement of "at most" 47.

In a binomial distribution, .

While there is no built-in command for "at least",

you can quickly find the result by creating this

complement situation by subtracting from 1. Just

remember to adjust the value to 47.

The adjusted formula for "at least" is 1 - binomcdf (n,

p, r - 1).Here is our answer to part c.

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The fact that this answer is the same as the "at most" answer for the number 52, is due to the symmetric nature of the distribution about its mean of 50.

Practice (independent odd # HW even #)

1. The probability that Kyla will score above a 90 on a

mathematics test is 4/5. What is the probability that she will

score above a 90 on exactly three of the four tests this

quarter?

Choose:

2. Which fraction represents the probability of obtaining exactly eight heads in ten

tosses of a fair coin?

Choose:

45/1024

64/1024

90/1024

180/1024

3. Experience has shown that 1/200 of all CDs

produced by a certain machine are defective. If a

Answer

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quality control technician randomly tests twenty

CDs, compute each of the following probabilities:

• P(exactly one is defective)

• P(half are defective)

• P( no more than two are defective)

4. A fair coin is tossed 5 times. What is the probability that it lands tails up exactly 3

times?

Choose:

5. After studying a couple's family history, a doctor determines that the

probability of any child born to this couple having a gene for disease X is

1 out of 4. If the couple has three children, what is the probability that

exactly two of the children have the gene for disease X?

Answer

6. If a binomial experiment has seven trials in which the

probability of success is p and the probability of failure is q,

write an expression that could be used to compute each of the

following probabilities:

• P(exactly five successes)

• P(at least five successes)

• P(at most five successes)

Answer

7. On any given day, the probability that the entire Watson

family eats dinner together is 2/5. Find the probability

that, during any 7-day period, the Watson's each dinner

together at least six times.

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Answer

8. When Joe bowls, he can get a strike (knock down all

of the pins) 60% of the time. How many times more

likely is it for Joe to bowl at least three strikes out of

four times as it is for him to bowl zero strikcs out of

four tries? Round answer to the nearest whole

number.

Answer

9. A board game has a spinner on a circle that has five equal

sectors, numbered 1, 2, 3, 4, and 5, respectively. If a player has

four spins, find the probability that the player spins an even

number no more than two times on those four spins.

Answer

10. Give an example of an experiment where it is appropriate to

use a normal distribution as an approximation for a binomial

probability. Explain why in this example an approximation

of the probability is a better approach than finding the exact

probability.

Answer

Key

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3. The 99% implies a distribution within 3 standard deviations of the mean. The difference from 585 milliliters to 595 milliliters is 10 milliliters. Symmetrically divided, there are 5 milliliters used to create 3 standard deviations on one side of the mean. Dividing 5 by 3, we get the standard deviation to be 1.67 milliliters, to the nearest hundredth.

4. Snowfall: 8.6, 9.5, 14.1, 11.5, 7.0, 8.4, 9.0, 6.7, 21.5,

7.7, 6.8, 6.1, 8.5, 14.4, 6.1, 8.0, 9.2, 7.1

The mean = 9.46 inches.

Standard deviation = 3.74

5. Scores: 100, 85, 55, 95, 75, 100.

From the graphing calculator we have the mean = 85 and

the population standard deviation = 16.07275127 = 16.

85 + 16 = 101 and 85 - 16 = 69

All but one of the scores falls in the range from 69 to 101.

Answer: 5 scores

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8. Scores: 276, 279, 279, 277, 278, 278, 280, 282, 285,

272, 279, and 278.

Mean = 278.5833333

Sample Standard Deviation =

3.146667309

One standard deviation above and below the mean creates a

range from 275.436666 to 281.7300006

(275 < score < 281). Of the 12 scores, 9 fall into this range,

which is 75% of the scores.

7. Use your graphing calculator. Be sure to remember that

you are working with grouped data (frequency table).

Mean = 3.6

Standard deviation = 2.9

One standard deviation from the mean

creates a range from 0.7 to 6.5 which

includes presidents with from 1 to 6

children.

There are a total of 31 such presidents.

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9. Mean = 11 and standard deviation = 1.5

a. 50% In a normal distribution, the mean divides the data into two equal areas. Since 11 is

the mean, 50% of the data is above 11 and 50% is below 11.

b. 12.5 is exactly one standard deviation above the mean. Examining the normal

distribution chart shows that 15.9%will fall above one standard deviation. Probability

is .159.

c. 12.5 and 8 are exactly one standard deviation above the mean and 2 standard deviations

below the mean respectively. Using the chart we

know:

10. 500 scores, mean 125, standard deviation 10 a. What percent of the values is in the interval 115 - 135?

mean + one standard deviation = 135

mean - one standard deviation = 115

15% + 19.1% + 19.1% + 15% = 68.2% (from chart)

Percent within one standard deviation of the mean = 68.2% = 68%

b. What percent of the values is in the interval 100 - 150?

mean + 2.5 standard deviations = 150

mean - 2.5 standard deviations = 100

2(1.7% + 4.4% + 9.2% + 15% + 19.1%) = 98.8% (from chart)

Percent with 2.5 standard deviations of the mean = 98.8% = 99%

c. What interval about the mean includes approximately 95% of the data? 2 standard

deviations about the mean for a total interval size of 40, with the mean in the center.

mean + 2 standard deviations = 145

mean - 2 standard deviations = 105

Interval: [105,145]

d. What interval about the mean includes 50% of the data?

.50% of the distribution lies within 0.67448 standard deviations about the mean for a

total interquartile range (size) of 13.4896, with the mean in the center.

mean + 0.67448 standard deviations = 131.7448

mean - 0.67448 standard deviations = 118.22552

Interval: [118.2552, 131.7448]

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Binomial

3. a.) P(exactly one is defective)

b.) P(half are defective) = P(exactly 10 are defective)

c.) P( no more than two are defective) = P( at most two are defective). None, one, or two

could be defective.

Sum = .9998662997 which is approximately 1 or 100%.

5.

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6.

a.)

b.)

c.)

7. At least means 6 times or 7 times.

8. At least three strikes:

Exactly zero strikes:

19 times more likely.

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9. There are 2 even numbers out of the 5 numbers.

No more than = at most.

10.

One possible answer: Find the probability of

getting at most 250 heads when flipping a fair

coin 300 times.

When dealing with very large samples, it can

become very tedious to compute certain

probabilities. In such cases, the normal

distribution can be used to more quickly

approximate the probabilities that otherwise

would have been obtained through laborious

computations.