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UGA - STAT 2000 - Class Notes - Week 7

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Tuesday, February 14th

Chapter 6: Probability Distributions

6.1 Summarizing Possible Outcomes and Their Possibilities

• Probability distribution: tells you all possible values and how their probabilities are distributed

o Ex: 60% of people have brown eyes, 25% of people have blue eyes, and 15% of people have green eyes (percentages probably aren’t accurate)

Probabilities of Possible Outcomes

• Probability distribution of a discrete random variables assigns a probability to each possible value of that random variable

• Two requirements for a discrete probability distribution

o All probabilities are between 0 and 1

o If you add all of the probabilities, you’ll get 1

• Example 1: Which are valid probability distributions?

Let X = # of red

lights when 3

traffic lights are

encountered

X

P(X)

0

0.2

1

0.3

2

0.1

3

0.4

Let X = number on

a die

X

P(X)

1

1/3

2

1/6

3

1/6

4

1/6

5

0

6

1/6

Let X = number on a

die

X

P(X)

1

1/2

2

-1/2

3

1/6

4

2

5

-1

6

-1/5

1. What probability distributions are invalid and why? The third one is invalid because it has negative numbers. Don't forget about the age old question of What are the types of risk?

The Mean of a Probability Distribution (…of a Discrete Random Variable)

• Many playoff and championship games in professional sports are decided by a best of seven series; first team to get four victories wins the series

• Consider following distribution where x = number of games needed to determine a winner in a baseball World Series when the chance of either team winning any game is 0.50. How many games are expected to play, on average, to determine a winner?

Number of Games, x

Probability P(x)

4

1/8 = 0.125

5

1/4 = 0.25

6

5/16 = 0.3125

7

5/16 = 0.3125

Mean for 10,000 games:

∝ = (1250x4) + (2500x5) + (3125x6) + (3125x7)

10000

Written slightly differently:

∝ = 4(0.125) + 5(0.25) + 6(0.3125) + 7(0.3125) = 0.5 + 1.25 + 1.875 + 2.1875 = 5.8125 • In general, the mean, ∝, or Expected Value, E(x) is If you want to learn more check out What is the most common lipid consumed by ruminants?

o E(x) = ∑ x⋅P(x)

• Expected value: a weighted average that represents what happens, on average, in the long run; not about one individual event

• Example 2: Lottery Ticket

Suppose a friend gives you a lottery ticket. There is a 1% chance that you win $1000 and a 5% chance that you win $100. Let X denote how much you win. Find E(X)

X

P(X)

1000

0.01

100

0.05

0

0.94

E(x) = 1000(0.01) + 100(0.05) + 0(0.94) = $15

Not probability that you will win $15

• Example 3: Grade Distribution

An instructor always assigns final grades such that 20% are A, 40% are B, 30% are C, and 10% are D. The grape point scores are 4 for A, 3 for B, 2 for C, and 1 for D. Find the mean of this probability distribution.

Mean = 4(0.2) + 3(0.4) + 2(0.3) + 1(0.1) = 0.8 + 1.2 + 0.6 + 0.1 = 2.7 We also discuss several other topics like What are the 4 P of marketing?

6.2 Probabilities for Bell-Shaped Distributions

• For continuous random variables, we will be looking at intervals instead of single values o Like when using the Empirical Rule

• Three normal curves

o Have different means and different standard deviations

o Flatter curves have larger standard deviations

o Curve on the right has the largest mean; curve on the left has the smallest • Standard normal distribution: mean of 0 and standard deviation of 1 o Z-scores follow normal distribution

• Area under a curve is equal to 1

o Ex: 95% of the area under curve is between 8 and 12

o Need calculus to determine the area under the curve but StatCrunch makes this easy Using StatCrunch to Find Probabilities

• Adult male heights normally distributed with a mean of 70 and standard deviation of 4 o For observation of 72, z-score is: (72-70)/4 = 0.5

• To access normal calculator, open StatCrunch then Stat ???? Calculators ???? Normal o Enter correct value in bottom left hand box, make sure inequality is flipped the correct direction, and click Compute We also discuss several other topics like what are the five themes of geography?

o Approximately 69% of adult males are 72 inches or lower

o Approximately 31% of adult males are 72 inches or higher

• Example 4: SAT Math Scores

Suppose SAT Math scores are normally distributed with a mean of 500 and a standard deviation of 100. If your SAT Math score is a 670, what percentage of SAT scores were higher than yours? Sketch a curve and shade the appropriate region. Find the z-score, then use StatCrunch to find the answer.

z-score = (670-500)/100 = 1.7

4.5% of SAT scores were higher than yours

• Example 5: Midterms

On a midterm exam, students who score between 80 and 90 receive a grade of a B. Suppose the scores on the exam have a normal distribution with mean 83 and standard deviation 12. What proportion of students get a B? Sketch a curve and shade the appropriate region. Then use StatCrunch to find the answer.

32% of students get a B

We also discuss several other topics like What is Information Processing theory?

• Example 6: Dart Weights

According to the National Dart Association, weight for tournament darts is nearly normal with mean 17 grams with standard deviation of 0.3 grams. For tournament play, all darts must be between 16.4 and 17.4 grams. What is the probability that a randomly selected tournament dart will be outside the range of acceptable weights?

11% of darts will be outside acceptable weight range

Use complement rule

Thursday, February 16th

Using Percentiles/Areas with the Normal Calculator

• We can use StatCrunch to figure out percentiles

o Plug the percentile (as a proportion) in the lower right hand box and change the inequality

o The pth percentile tells you that p% of the data are below that number • When using the normal calculator:

o Make sure to use the correct mean and standard deviation

o For some problems, you may need to change the direction of the inequality o Input variable values into the bottom left box, input percentiles/areas into the bottom right box

• Example 7: Running

Suppose the number of miles you run per week is approximately normally distributed with a mean of 9.5 and standard deviation of 1.5. (use StatCrunch) If you want to learn more check out What causes abdominal aorta?

a. How often do you run less than 7 miles per week? (What is probability in any one randomly selected week that you ran less than 7 miles?)

0.048 or 4.8%

b. How often do you run between 8 and 12 miles per week?

0.79 or 79%

c. How many miles represents the highest 90% of miles run?

7.58 miles

• Example 8: Wii Fit Study

A Wii Fit study tested men and women between the ages of 20-24 years old and found that the mean amount of calories burned after 30 minutes on the Super Hula Hoops was 111 calories. Suppose the distribution of calories burned is bell-shaped and the standard deviation is 15 calories.

a. A 22-year-old woman burns 130 calories on the Super Hula Hoop. What is her z score? What is her percentile?

1.2667; 0.8974

b. How many calories would a man or woman have to burn on the Super Hula Hoop in 30 minutes to be at the 65th percentile?

116.8

Connections to Calculus

• Note: You don’t have to know any of the following information. Just like any other curve, the normal curve as an equation:

• When we say that the area under the curve equals 1, what we are actually saying is:

• That is, if we were to integrate this function from –∞ to ∞, we’d get 1

6.3 Probabilities When Each Observation Has Two Possible Outcomes

• About 45% of people have type O blood. Suppose you select 3 people at random and check their blood type. What is the probability that all 3 have type O blood? What is the probability that 2 have type O blood? What is the probability that only 1 has type O blood? What is the probability that none have type O blood?

S = {OOO, OON, ONO, NOO, ONN, NON, NNO, NNN}

P(3 type O) = P(OOO) = (0.45)(0.45)(0.45) = 0.0911

P(2 type O) = P(OON or ONO or NOO) = 3(0.45)(0.45)(0.55) = 0.3341

P(1 type O) = P(ONN or NON or NNO) = 3(0.45)(0.55(0.55) = 0.4084

P(no type O) = P(NNN) = 1(0.55)(0.55)(0.55) = 0.1664

• Binomial distribution: a special discrete probability distribution

o Each of n trials has two possible outcomes. The outcome of interest is called a success and the other outcome is called a failure.

o Each trial has the same probability of success. This is denoted by p, so the probability of a success is p and the probability of a failure is q = 1 – p.

o The n trials are independent. That is, the result for one trial does not depend on the results of other trials (the outcome of one trial does not influence the outcome of another trial).

• Notation for a binomial experiment:

o n = number of trials (this is fixed and known)

o p = probability of success for any given trial

o q = probability of failure for any given trial (q = 1 – p)

o X = number of successes for n trials

o ***n and p are the parameters of the binomial random variable***

• Example 9: High School Graduation Rates

The U.S. high school graduation rate was 68.8% in 2007.

a. Explain why the binomial distribution is the appropriate model.

Two outcomes: high graduate or not

n = 25

p = 0.688

Independence? Yes, from random sampling

b. What is the probability that more than 15 students graduated?

P(x > 15) = 0.772

• Example 10: Drinking and Driving

Based on a study of that followed 1,250 first-year college students enrolled at a large mid Atlantic university, 1 out of 5 college students admit to drinking and driving. Assume that

the 1250 students are representative of all American college students. Fifteen college students are randomly selected. (Two outcomes: yes or no; n = 15; p = 0.2) Find the probability that:

a. Exactly 5 have driven after drinking

0.103

b. Less than 5 have driven after drinking

0.836

c. At most 5 have driven after drinking

0.939

• Example 11: Commuter Airline

A commuter airline has 8 seats. Passengers having reservations show up independently. The probability of each passenger showing up is 0.80. If the airline accepts 10 reservations for the 8-seat plane, what is the probability that all passengers who show up can’t be accommodated?

Two outcomes: passenger shows up or not

Independent: yes, told this in question

n = 10; p = 0.8

P(x = 9) + P(x = 10) = 0.376

Expected value: np = 10 x 0.80 = 8

• Mean and standard deviation for a binomial random variable:

o ∝ = np

o σ = ���