Outline

Lecture 6

1. Two kinds of random variablesa. Discrete random variablesb. Continuous random variables

2. Symmetric distributions3. Normal distributions4. The standard normal distribution5. Using the standard normal distribution6. The normal approximation to the binomial

1. Two kinds of random variables

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a. Discrete (DRV) Outcomes have countable values

Possible values can be listed

E.g., # of people in this room Possible values can

be listed: might be …51 or 52 or 53…

1. Two kinds of random variables

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a. Discrete RVb. Continuous RV

Not countable Consists of points in

an interval E.g., time till coffee

break

1. Two kinds of random variables

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The form of the probability distribution for a CRV is a smooth curve.

Such a distribution may also be called a

Frequency Distribution

Probability Density Function

1. Two kinds of random variables

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In the graph of a CRV, the X axis is whatever you are measuring

E.g., exam scores, mood scores, # of widgets produced per hour.

The Y axis measures the frequency of scores.

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X

The Y-axis measures frequency. It is usually not shown.

2. Symmetric Distributions

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In a symmetric CRV, 50% of the area under the curve is in each half of the distribution.

P(x ≤ ) = P(x ≥ ) = .5

Note: Because points on a CRV are infinitely thin, we can only measure the probability of intervals of X values

We can’t measure or compute the probability of individual X values.

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A symmetric distribution which is not mound-shaped. The two sections (above and below the mean) each contain 50% of the observations.

μ

50% of area on each side of µ

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µ

50% of area50% of area

A mound-shaped symmetric distribution (the normal distribution)

3. Normal Distributions

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A very important set of CRVs has mound-shaped and symmetric probability distributions. These are called “normal distributions”

Many naturally-occurring variables are normally distributed.

3. Normal Distributions

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Are perfectly symmetrical around their mean, .

Have standard deviation, , which measures the “spread” of a distribution

is an index of variability around the mean.

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µ

3. Normal distributions

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There are an infinite number of normal distributions

They are distinguished on the basis of their mean (µ) and standard deviation (σ)

3. Standard Normal Distribution

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The standard normal distribution is a special one produced by converting raw scores into Z scores

Thus, µ = 0 and σ = 1

3. Standard Normal Distribution

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The area under the curve between and some value X ≥ has been calculated for the standard normal distribution and is given in Table IV of the text.

E.g., for Z = 1.62, area = .4474

Because distribution is symmetric, table values can also be used for scores below the mean (Z scores below 0).

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XZ = 1.62Z = 0

Area gives the probability of finding a score between the mean and X when you make an observation

.4474

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XZ = -1.62

For Z < 0, same values can be used as for Z > 0

.4474

5. Using the Standard Normal Distribution

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Suppose the average height for Canadian women is µ = 160 cm, with = 15 cm.

What is the probability that the next Canadian woman we meet is more than 175 cm tall?

Note two things:

1. this is a question about a single case

2. it specifies an interval.

Using the Standard Normal Distribution

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160 175

We need this areaTable gives this area

cm

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Remember that area above the mean, , is half (.5) of the distribution.

µ

Using the Standard Normal Distribution

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160 175

Call this shaded area P. We can get P from Table IV

Using the Standard Normal Distribution

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Z = X - = 175-160

15= 1.00

Now, look up Z = 1.00 in the table.

Corresponding area is .3413.

• Value of Z that marks one end of the interval – you want to find probability that a randomly selected case has a score that falls in this interval

• Other end of the interval is at µ (= 0)

µ Z

Using the Standard Normal Distribution

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160 175

This area is .3413 So this area must be .5 – .3413 = .1587

Using the Standard Normal Distribution

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Z = 0 Z = 1.0

This area is .3413

So this area must be .5 – .3413 = .1587

Using the Standard Normal Distribution

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What is the probability that the next Canadian woman we meet is more than 175 cm tall?

Answer: .1587

Binomial Random Variable – Method #3

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When n is large and p is not too close to 0 or 1, we can use the normal approximation to the binomial probability distribution to work out binomial probabilities.

How can you tell if the normal approximation is appropriate in a given case?

Use the approximation if np ≥ 5 and nq ≥ 5

BRV – Method #3

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Histogram Normal curve

XThe histogram shows the probabilities of different values of the BRV X. Because area gives probability, we can use the area under a section of the normal curve to approximate the area of some part of the histogram

BRV – Method #3

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In order to use the normal approximation, we have to be able to compute the mean and standard deviation for the BRV (in order to work out Z).

μ = np (# of observations times P(Success))σ = √npq

There’s just one other issue to deal with…

BRV – Method #3

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X1 2 3 4 5 6 7 8 9 10

Notice how the normal curve misses one top corner of the rectangle for 7 and overstates the other top corner – these two errors approximately cancel each other.

BRV – Method #3

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X1 2 3 4 5 6 7 8 9 10

Notice how the rectangle for 7 runs from 6.5 to 7.5. The probability of X values up to and including 7 is given by the area to the left of 7.5.

BRV – Example 1 from last week

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Air Canada keeps telling us that arrival and departure times at Pearson International are improving. Right now, the statistics show that 60% of the Air Canada planes coming into Pearson do arrive on time. (This actually is an improvement over 10 years ago when only 42% of the Air Canada planes arrived on time at Pearson.) The problem is that when a plane arrives on time, it often has to circle the airport because there’s still a plane in its gate (a plane which didn’t leave on time). Statistics also show that 50% of the planes that arrive on time have to circle at least once, while only 35% of the planes that arrive late have to circle at least once.

BRV – Example 1 from last week

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c) Of the next 80 Air Canada planes that arrive at Pearson, what’s the probability that between 40 and 45 (inclusive) have to circle at least once?

BRV – Example 1

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First, we check to see whether we can use the normal approximation. To do this, we need to know the probability that a plane has to circle at least once:

P(C) = P(C ∩ Late) + P(C ∩ Not Late) = [P(L) * P(C │L)] + [P(L) * P(C│L)] = .35 (.40) + .6 (.5) = .44

BRV – Example 1

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Now we can do the check:

n = 80. p = .44 and q = (1 – p) = .56

np = 80 (.44) = 35.2 > 5nq = 80 (.56) = 44.8 > 5

Thus, it’s OK to use the normal approximation.

BRV – Example 1

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μ = np = 80*(.44) = 35.2σ = √npq = √80(.44)(.56) = 4.44

Correction for continuity: to get area for 40 and up, we use X = 39.5. To get area for 45 and below, we use X = 45.5

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35.2 40 45

40 and up starts at 39.5

45 and below starts at 45.5

BRV – Example 1

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Z39.5 = 39.5 – 35.2 = +.97

4.44

Z45.5 = 45.5 – 35.2 = +2.32

4.44

P(.97 ≤ Z ≤ 2.32) = .4894 - .3340 = .1554.

This is the probability that between 40 and 45 (inclusive) of the next 80 planes have to circle at least once.

Probability that Z ≤ 2.32

Probability that Z ≤ .97

35.2 40 45

From here down = .5 + .3340 = .8340

From here down = .5 + .4894 = .9894

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35.2 40 45

Using the normal approximation, we estimate the combined area of all these rectangles to be .9894 – .8340 = .1554

CRV Example 1

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Wind speed in Windy City is normally-distributed and the middle 40% of wind speeds is bounded by 23.9 and 29.3.

a. Wind speed would be expected to be lower than what value only 5% of the time?

23.9 µ 29.3

.40

.20

Since distribution is symmetric, µ = midpoint between 23.9 and 29.3. This is 26.6.

What value of Z is associated with p = .20?

From Table, Z = 0.53

CRV Example 1a

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Since Z = X - µ then, σ = X - µ

σ = 29.3 – 26.6 = 5.0940.53

Z for p = .45 is 1.645 (from Table)

Thus, required X = 26.6 – 1.645 (5.094) = 18.22

σ Z

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.05

.45

µ18.22

CRV Example 1

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Wind speed in Windy City is normally-distributed and the middle 40% of wind speeds is bounded by 23.9 and 29.3.

b. UV radiation in Windy City is normally distributed with a mean of 10.4 and a variance of 6.25. Wind speed and UV radiation are independent of each other. A "bad day" in Windy City is any day on which either wind speed exceeds 35.0 or UV radiation exceeds 15. What is the probability of a bad day in Windy City?

CRV Example 1b

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√6.25 = 2.5

10.4 15

CRV Example 1b

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Z = 15 – 10.4 = 1.84 2.5

P(Z < 1.84) = .4671 (from Table)

P(X > 15) = P(Z > 1.84) = .5 – .4671 = .0329

CRV Example 1b

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5.094

26.6 35

CRV Example 1b

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Z = 35 – 26.6 = 1.655.094

P(Z < 1.65) = .4505 (from Table)

P(X > 35) = P(Z > 1.65) = .5 – .4505 = .0495

CRV Example 1b

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P(Bad Day) = P[(UV > 15) or (Wind > 35)]

= .0329 + .0495 – (.0329)(.0495)

= .0808

(From Additive Rule of Probability)

CRV Example 2

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Three truckers from different companies meet at a truck-stop and start talking about how fast their companies are at getting freight to its destination. Trucker #1 tells the others that Acme, his company, requires drivers to make the Toronto-Quebec City run in 726 minutes, though the actual mean duration for the trip is 735, with a standard deviation of 9 minutes. Trucker #2 reveals that Road Hog Transport, his company, wants drivers to make the same run in 732 minutes, but the actual mean time is 740 minutes, with a standard deviation of 12. Trucker #3 claims that his company, TakeAChance Transport, gives drivers 754 minutes to make the run, and that the actual mean trip length is 760 minutes with a standard deviation of 18. Assume that trip times are normally distributed.

CRV Example 2

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a. The three drivers each make the Toronto-Quebec City run, leaving the truck-stop at the same moment. Acme’s driver makes the run in the time specified by his company. What is the probability that he beats the Road Hog driver?

CRV Example 2

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Z = 726 – 740 = –14 = – 1.167 12 12

P(Z ≤ – 1.167) = .3790.

Thus, probability that Acme beats Road Hog is .3790 + .5 = .8790.

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726 740

0.50.3790

CRV Example 2

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b. What is the probability that, relative to the time specified by their respective companies, all three drivers arrive at least 20 minutes late?

CRV Example 2

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ZACME = 746 – 735 = 1.229

P(Z ≥ 1.22) = .5 - .3888 = .1112

ZROAD HOG = 752 – 740 = 1.00 12

P(Z ≥ 1.00) = .5 - .3413 = .1587

ZTAKEACHANCE = 774 – 760 = .777 P(Z ≥ .777) = .2177

18

CRV Example 2

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c. Suppose Trucker #2 arrived 20 minutes lat and Trucker #3 arrived 30 minutes early (relative to their respective companies’ demands). What is the probability that Trucker #1 was the second to arrive?

CRV Example 2

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Z1 = 752 – 735 = 17 = 1.889 9 9

Z2 = 724 – 735 = 11 = –1.222 9 9

P(Z ≤ 1.889) = .4706P(Z ≥ –1.222) = .3888

.8594Thus, P (Trucker #1 arrives second) = .8594

Review

Lecture 6

Area under curve gives probability of finding X in a given interval.

Area under the curve for Standard Normal Distribution is given in Table IV.

For area under the curve for other normally-distributed variables first compute:

Z = X -

Then look up Z in Table IV.