Chapter 7 Student HandoutProbability Distributions: Binomial, Geometric, and Normal Distributions

Assignments:a. 1, 2, 7, 9, 11, 16, 19b. 20, 23, 27, 28, 32, 37, 40c. 45, 47, 53, 56, 62, 63

Test Ad. 64, 67, 69, 73, 76, 80e. 82, 86, 87, 88, 92f. 93, 97, 101g. Rotten Eggs activity

Test B

Do you know???1. What is a random variable?2. What is the difference between a discrete random variable and a continuous one?3. What do the probabilities add up to in a probability distribution?4. How do you find the expected value of a discrete probability distribution by hand? How

do you find the standard deviation by hand?5. How do you find probabilities from the graph of a probability distribution?6. What is a density curve/histogram?7. What is required for a random variable to have a binomial distribution?8. Give an example of a binomial experiment. Show how it meets the criteria.9. What is required for a random variable to have a geometric distribution?10. Give an example of a geometric experiment. Show how it meets the criteria.11. Under what circumstances are trials involving sampling without replacement allowed for

binomial distributions?12. Do binomial and geometric distributions involve continuous or discrete random variables?13. What is the formula for finding P(X=x) if X has a binomial distribution?14. How do you find cumulative probabilities for binomial distributions? Does it make a difference

whether it is “or equal to” or not?15. How do you find the mean of a binomial distribution?16. How do you find the expected value of a binomial distribution?17. How do you find the standard deviation of a binomial distribution? 18. When combining to random variables, X and Y, by addition, what is the mean of X+Y? What is

the standard deviation of X+Y? Answer these questions for subtraction.19. When you multiply a random variable by a constant, like changing units from inches to

centimeters, what happens to the mean and standard deviation of the random variable?20. When you add a constant to a random variable what happens to the mean and the standard

deviation?

End of part A.

21. Is a normal distribution discrete or continuous?22. What are the mean and standard deviation of the standard normal distribution?23. How many standard deviations wide is the standard normal distribution?24. What is the z-score formula? What do z-scores tell you?25. How many standard deviations wide is a non-standard normal distribution?26. How do you find P(X<x) given that X has a normal distribution with a certain mean and a certain

standard deviation? Does “or equal to” make a difference?

27. How do you find x if you know X has a normal distribution and you know either the greater than or less than probability associated with x?

28. Why can’t you do “equal to” probabilities for normal distributions?29. How do check for normality of data?30. How do you normalize right skewed data?31. How do you normalize left skewed data?32. When can you use a normal distribution to approximate a binomial distribution?33. How do you correct for continuity when using a normal distribution to approximate a binomial

distribution?End of part B.

Connections…Chapters 3 & 4 - Used statistics to determine “what happened”.

Chapter 6 - Studied the basic principles of probability.

Chapter 7 - Use a basic understanding of the circumstances involved to construct a probability distribution of the relative frequencies we expect to result in a given situation.

Concepts…Random Variables - (we don’t know what they equal before hand)

A variable that has a single numerical value (determined by chance) for each outcome of an experiment.

Discrete Random Variable - Ex: number of red shirts in a sporting goods store Continuous Random Variable - the temperature of a stovetop burner set on high

Probability Distribution - gives the probability for each value of a random variable

Representing a Probability Distribution - A probability histogram created from a table based on probabilities rather than actual relative frequencies. (Note: the sum of all P(x) must equal 1 and each P(x) must be between 0 and 1, self inclusive)

Mean, Variance, and Standard Deviation for a Probability Distribution – use formulas or the calculator

Expected Value = the Mean of the Probability Distribution

EXPECTED VALUE AND STANDARD DEVIATION OF RANDOM VARIABLES

Random Variables:A discrete random variable X has a countable number of possible values. The probability distribution of X lists the values and their probabilities.

Value of X x1 x2 x3 … xn

Probability p1 p2 p3 … pn

Where 0≤pi≤1 for all iAndp1 + p2 + p3 + … + pn = 1.

On a 5-item multiple choice exam there are choices A-D on each item. A student who blindly guesses on each item may get anywhere from 0 to 5 correct. Let X = the number correct. The probability distribution of X is…

Value of X 0 1 2 3 4 5Probability .23730 .39551 .26367 .09789 .01465 .00098

Verify that this is a legitimate probability distribution for this random variable.

Find the probability that X is 2 or 3. Use correct notation.

Find the probability that X is greater than 3. Use correct notation.

Find the probability that the student gets at least 1 correct. Use correct notation.

Means and Variances of Random Variables:

Suppose the probability distribution of X is

Value of X x1 x2 x3 … xn

Probability p1 p2 p3 … pn

Then…

I do believe in ferries, I do, I do!Consider a small ferry that can accommodate cars and buses. The toll for cars is $3 and the toll for buses is $10. Let x and y denote the number of cars and buses, respectively, carried on a single trip. Suppose that x and y are independent and have the probability distribution below.

X 0 1 2 3 4 5P(X) .05 .10 .25 .30 .20 .10Y 0 1 2P(Y) .50 .30 .20

a) Compute the mean and variance of x.b) Compute the mean and variance of y.c) Compute the mean and variance of the total amount of money collected in the tolls from

cars.d) Compute the mean and variance of the total amount of money collected in the tolls from

buses.e) Compute the mean and variance of z = total number of vehicles (cars and buses) on the

ferry.Compute the mean and variance of w = total amount of money collected in tolls.

The mean of X is E( X )=μx=∑ x i⋅pi

And the variance of X isσ

x 2=∑ [ ( xi−μx )2⋅pi ]

So the standard deviation of X isσ x=√σ X

2

Review:

Define a Random Variable: ______________________________________________________

______________________________________________________________________

Define Independent Random Variables: ____________________________________________

______________________________________________________________________

What is meant by Expected Value of a random variable? ______________________________

______________________________________________________________________

Give an example of why we would want to combine two random variables. _______________

_____________________________________________________________________

What are the rules for linear transformations of a random variable and for combining random variables?

1. The senior class is selling raffle tickets at the Senior Auction and Dinner night. Each raffle ticket costs $10 and they plan on selling 200 raffle tickets. The prizes, which have been paid for from the money collected from the sale, are one $100 stereo system, two $75 Dillard's gift certificates, four $30 Chili's gift certificates, and eight $15 AMC gift certificates. What is the expected value of a raffle ticket? How much profit per raffle ticket is the senior class expecting to make?

2. A study of how well sophomores adapt the first six weeks of the school year, 50 randomly selected 10th graders office referral records were examined to determine how often the tenth graders needed administrative guidance to a positive road to follow on campus.# of referrals, x 0 1 2 3 4 5 6+P(X = x) .46 .22 .12 .10 .06 .02 .02

What is the expected number of referrals for MHS 10th graders? E(X) = ________

What is the standard deviation of number of referrals? x = _________

3. Locate the formulas for Expected Value and Standard Deviation of a Random Variable on the AP Statistics formula chart.

Also remember…Stat…CALC…1-Var Stats L1, L2…x̄ is the expected value and σ is the standard deviation.

Expected Value – Multiple Choice problems

1) Cans of soft drinks cost $.30 in a certain vending machine. What is the expected value and variance of daily revenue (Y) from the machine, if X, the number of cans sold per day has E(X) = 125, and Var(X) = 50 ? a) E(Y) = 37.5 Var(Y) = 50 b) E(Y) = 37.5 Var(Y) = 4.5 c) E(Y) = 37.5 Var(Y) = 15 d) E(Y) = 37.5 Var(Y) = 15 e) E(Y) = 125 Var(Y) = 4.5

2) A crop insurance company establishes the following loss table based upon previous claims percent loss | 0 25 50 100probability | .90 .05 .02 ????

If they write policy that pays a maximum of $150/hectare, their expected loss in $/hectare is approximately:

a) 5.2 b) 7.9 c) 4.5 d) 37.5 e) 25.0

3) A rock concert producer has scheduled an outdoor concert. If it is warm that day, she expects to make a $20,000 profit. If it is cool that day, she expects to make a $5,000 profit. If it is very cold that day, she expects to suffer a $12,000 loss. Based upon historical records, the weather office has estimated the chances of a warm day to be .60; the chances of a cool day to be .25. What is the producer's expected profit? a) $5,000 b) $13,000 c) $15,050 d) $13,250 e) $11,450

4) A restaurant manager is considering a new location for her restaurant. The projected annual cash flow for the new location is:

Annual Cash Flow $10,000 $30,000 $70,000 $90,000 $100,000Probability 0.10 0.15 0.50 0.15 ?

The expected cash flow for the new location is:

a) $12,800 b) $64,000 c) $70,000 d) $60,000 e) $50,000

5) An insurance company has estimated the following cost probabilities for the next year on a particular model of car:

cost | $0 $500 $1000 $2000prob | .60 .05 .13 ????

The expected cost to the insurance company is (approximately):

a) $155 b) $595 c) $875 d) $645 e) $495

6) Before planting a crop for the next year, a producer does a risk assessment. According to her assessment, she concludes that there are three possible net outcomes: a $7,000 gain, a $4,000 gain, or a $10,000 loss with probabilities 0.55, 0.20 and 0.25 respectively. The expected profit is: a) $3,850 b) $0 c) $2,150 d) $2,500 e) $800

7) A business evaluates a proposed venture as follows. It stands to make a profit of $10,000 with probability 3/20, to make a profit of $5,000 with probability 9/20, to break even with probability 1/4 and to lose $5,000 with probability 3/20. The expected profit in dollars is: a) 1,500 b) 0 c) 3,000 d) 3,250 e) 1,500

8) The average length of stay in a hospital is useful for planning purposes. Suppose that the following is the distribution of the length of stay in a hospital after a minor operation:

Days 2 3 4 5 6Prob .05 .20 .40 .20 ?

The average length of stay is:

a) .15 b) .17 c) 3.3 d) 4.0 e) 4.2

9) An insurance company issues a policy on a small boat under the following conditions: The replacement cost ($5000) will be paid for a total loss. If it is not a total loss, but the damage is more than $2000, then $1500 will be paid. Nothing will be paid for damage costing $2000 or less and of course nothing is paid out if there is no damage. The company estimates the probability of the first three events as .02, .10, and .30 respectively. The amount the company should charge if it wishes to make a profit of $50 above the expected amount paid out in a year is: a) $250 b) $201 c) $300 d) $1200 e) $165

Binomial and Geometric Probability

Binomial Properties Geometric Properties1. Fixed number of trials, n.2. Only two mutually exclusive outcomes.3. Independent trials.4. Probability of outcome is constant for each trial,

.

1. Only two mutually exclusive outcomes.

2. Independent trials.3. Probability of outcome is constant for

all trials, .

X is binomial random variable.

P( X=k )= n!k !(n−k )!

(π )k (1−π )n−k

where k is the number of successes (or failures)

X is geometric random variable.

P( X=a )=π (1−π )a−1

where a is the number of attempts to a success (or failure)

Mean Value (Expected Value)

E( X )=μx=nπ

Standard Deviation

σ x=√nπ (1−π )

Mean Value (Expected Value)

E( X )=μx=

1π

Standard Deviation

σ x=√ 1−π

π2

TI-84For a single event:P(X = k) binompdf (n, , k)

For cumulative events:P(X k) binomcdf (n, , k)

TI-84 For a single event: P(X = a) geometpdf (, a)

For cumulative events: P(X a) geometcdf (, a)

Example: Assume 40% of the market prefers Pepsi. What is the probability that 5 out of 10 randomly selected cola drinkers would choice Pepsi in a blind taste test?

P( X=5)=10!5 !(10−5 )!

( . 4 )5 (1−. 4 )10−5

= .20066

Example: Assume 40% of the market prefers Pepsi. What is the probability that the first cola drinker to randomly select Pepsi in a blind taste test would be the fifth customer?

P( X=5)=. 4 (1−. 4 )5−1=. 05184

Example: Assume 40% of the market prefers Pepsi. What is the probability that 5 or 6 out of 10 randomly selected cola drinkers would choice Pepsi in a blind taste test?

P(X = 5 or 6) = P(X = 5) + P(X = 6) = ?

use right to scroll

add .20066 + .11148 OR

Example: Assume 40% of the market prefers Pepsi. What is the probability that the first cola drinker to randomly select Pepsi in a blind taste test would be the fifth or sixth customer?

P(X = 5 or 6) = P(X = 5) + P(X = 6) = ?

add .05184 + .031104 OR

Example: Suppose the likelihood of Barry Bonds hitting a homerun is .2, what is the probability he will hit ten or fewer homeruns out of 30 attempts.

P(X 10) = ?

Example: Suppose the likelihood of Barry Bonds hitting a homerun is .2, what is the probability he will hit his first homerun within his first ten times at bat?

P(X 10) = ?

Example: Suppose the likelihood of Barry Bonds hitting a homerun is .2, what is the probability he will hit ten or more homeruns out of 30 attempts.

P(X 10) = 1 – P(X 9) = ?

Example: Suppose the likelihood of Barry Bonds hitting a homerun is .2, what is the probability he will hit his first homerun after his first ten times at bat?

P(X > 10) = 1 – P(X 10) = ?

Binomial and Geometric Probability Worksheet

Directions: Use proper probability notation to transfer from the verbal question. Then select distribution function on calculator and enter appropriate values. Write numerical answer following probability statement.

1. On the SAT, there are five answer choices (A, B, C, D, and E). The probability of randomly guessing the correct answer is .2.

a) What is the probability that on a 25-question section of the SAT by complete random guessing that exactly 8 questions will be answered correctly?

b) What is the probability that on a 25-question section of the SAT by complete random guessing that 6 or fewer questions will be answered correctly?

c) What is the probability that on a 25-question section of the SAT by complete random guessing that the first correctly guessed answered is the fourth?

d) What is the probability that on a 25-question section of the SAT by complete random guessing that the first correct answer will be within the first 6 guesses?

e) What is the expected number of correct guesses on a 25-question section of the SAT exam?

2. Major universities claim that 72% of their senior athletes graduate that year. Fifty senior athletic students attending major universities are randomly selected and recorded in order of selection.

a) What is the probability that exactly 40 senior athletic students graduate that year?b) What is the probability that 40 or 41 or 42 senior athletic students graduated that

year?c) What is the probability that 40 or fewer senior athletic students graduated that

year?d) What is the probability that 41 or more senior athletic students graduated that

year?e) What is the probability that 40 or more senior athletic students graduated that

year?f) What is the probability that the first senior athletic student to graduate in the

group of 50 that year is the 5th selected?g) What is the probability that the first senior athletic student to graduate in the

group of 50 that year is the 30th selected?h) What is the probability that the first senior athletic student to graduate in the

group of 50 that year is within the first 10 selected?i) What is the expected number of senior athletic students to graduate that year?j) What is the standard deviation of senior athletic students graduating that year?

3. Will Fumble is the only receiver for MHS football team with the likelihood of catching a pass of .15.

a) What is the probability that 2 passes are caught out of 6 passes?b) What is the probability that no passes are caught out of 6 passes?c) What is the probability that only 0 or 1 pass is caught out of 6 passes?d) What is the probability that 2 or fewer passes are caught out of 6 passes?e) What is the probability that more than 2 passes are caught out of 6 passes?f) What is the probability that the first pass caught is on the 1st pass?g) What is the probability that the first pass caught is on the 4th pass?h) What is the probability that the first pass is caught within the first 3 attempts?i) What is the probability that the first pass is caught after the first 3 attempts?j) What is the expect number of catches with 6 attempts?k) What is the expect number of attempts for the first pass caught?

Review:

Q: What are the qualities of a binomial experiment?

A: The experiment must have a fixed number of trials. The trials must be independent. Each trial must have all outcomes classified into two categories. The probabilities must remain constant for each trial.

Q: For Binomial Distributions what do p and x refer to?

A: p is the probability of the success of one, single, solitary, lone, individual, isolated, trial. x is the total number of successes in n trials.

Q: How do we define success for a Binomial Experiment?

A: It is an arbitrary designation not necessarily indicative of a desired result.

Q: What is the rule of thumb for considering sampling without replacement to be independent even though, strictly speaking, it is not.

A: When the sample size is no more than 5% of the population size.

Binomial Probability Formula

Binomial Probability Table

2nd =>DISTR=>binompdf(n, p, x)

Normal Probability Distributions

All have the same shape…unimodal and symmetrical…The Bell Curve!All are 6 standard deviations wide.All are centered on the mean.All obey the Empirical Rule.They are continuous distributions.Normal distributions can be standard or non-standard

What’s a Z-Score?Z-scores are standardized values based on the random variable’s value, population mean and standard deviation. X ~ N(, ) means that X is a random variable with a Normal distribution that has the given mean and the given standard deviation. If the random variable has a normal distribution, the distribution of the z-scores will also be normally distributed with a population mean of zero and standard deviation of 1, Z ~ N(0, 1).

The formula for relating the four unknowns, z, x, , and is z= x−μ

σ .Assuming X ~ N(66, 2), use the formula to calculate the corresponding z-scores for the x-values of 60, 62, 64, 66, 68, 70, and 72.

If we don’t know the values of x, but we know X ~ N(40, 4), then we can calculate the missing corresponding x-values when the z-score is –3, -2, -1, 0, 1, 2, and 3. Remember Z ~N(0, 1).

For the following problems, first sketch and label a normal distribution with all the information

provided. Then write the formula z= x−μ

σ . Substitute values of z, x, , and that are given and solve for the one value not given.

1. Suppose the average height of freshmen at MMHS is 60 inches with a standard deviation of 1.5 inches. What is the z-score for a freshman who has a height of

a) 58 inches?b) 60.15 inches?c) 56.25 inches?d) 70 inches?

2. Suppose the average height of sophomores at MHS is 62 inches with a standard deviation of 2 inches. What is the height of the sophomore (x-value) that corresponds to a

a) z-score = 0?b) z-score = -2.44?c) z-score = 1.76?d) z-score = 3.1?

3. Suppose the average height of juniors at MHS is unknown but the standard deviation is 2.5 inches. What is the population mean height of juniors if

a) a junior 66 inches tall corresponds to a z-score of -.75?b) a junior 71 inches tall corresponds to a z-score of 1.55? (This resulting population

mean should be different from the answer to a.)

4. Suppose the height of seniors at MHS is 67 inches but the standard deviation is unknown. What is the standard deviation knowing

a) a senior 68.5 inches tall has a corresponding z-score of .87?b) a senior 63 inches tall has a corresponding z-score of –2.43? (This resulting

population standard deviation should be different from the answer to a.)

Remember that there are four unknowns (z, x, , and ) in the formula z= x−μ

σ . You will be given three of the four values to calculate the last unknown.

Why do we calculate z-scores?

The z-score “marks” a spot to shade to the left or right under the normal curve Z ~ N(0, 1). The shaded area represents the likelihood (probability) of a sample statistic x to have been randomly drawn from a population distribution X~N(, ).

Example A: Suppose teachers at MHS have an age distribution X ~ N(40, 8). What is the likelihood that a randomly selected teacher from this population would have an age of 25 or younger?

z= x−μσ

z=25−408 z=−1 . 875

P(z -1.875) = 0.03036 (This can be looked up on the standard normal table or found with the calculator.)

So how likely is this?

Example B: Assuming the same distribution exists for age of teachers at MHS, how likely is it for a randomly selected teacher from MHS to be older than 50 years of age?

z= x−μσ

z=50−408 z=1 .25

P(z > 1.25) = 0.10562 (Table or calculator.)

Again, how likely is this?

Example C: Again using the same teacher age distribution at MHS, what is the probability that a randomly selected teacher’s age would fall somewhere between 30 and 50 years of age?

We know the z-score of 1.25 corresponds to teacher’s age of 50 as calculated in example B.

z= x−μσ

z=30−408 z = -1.25

P(-1.25 < z < 1.25) = 0.78870 P(z < 1.25) – P(z < -1.25) = 0.89432 – 0.10562 = 0.78870

Let’s try a few problems. Find the z –scores, show the proper notation, and draw and shade the normal curve.

Suppose the average teenage romantic relationship is normally distributed with a mean number of 100 days with a standard deviation of 30 days.

5. What is the likelihood that a randomly selected couple’s relationship has lasted for 90 days or fewer?

6. What is the probability that a randomly selected teenage bliss goes on less than 14 days?

7. How likely is it to randomly select a couple to find their admiration (relationship) has lasted 180 days or more?

8. What is the probability that a randomly selected love relationship has lasted more than one year (hint 365 days)?

9. What is the likelihood that a randomly selected couple’s relationship survived between 90 and 150 days?

10. How likely is it for teenage bliss to endure between 30 and 60 days?

Finding a z-score when we know probability

Example D: What is the z-score if the area shaded to the left under the normal curve with population mean of 0 and standard deviation of 1 is 0.25?

As we can see the z-score falls between –1 and 0, we can use TI-83+ to calculate the z-score for us. The function key we need is the invNorm command. This can be found by using the catalog or distribution commands. To use the invNorm enter the area shaded to the left in the normal curve, then a comma, the population mean (in this case, 0), a comma, and the standard deviation (in this case, 1).

Example E: What is the z-score if the area under the curve to the right is 84%, knowing the normal distribution has a mean of 0 and standard deviation of 1?

If the shaded area to the right represents 84%, then the non-shaded area to the left must represent the remaining 16% of the normal curve. The z-score separating the two sections can be found easier by using the area to the left with the technology of the TI-83+ calculator. Remember, the first number entered must represent the area to the left endpoint of the distribution.

Finding an x-value when we know probability

Example F: What is the value of a randomly selected value from a normal distribution with a population mean of 50 and standard deviation of 5, if the area shaded to the left represents 40% of the curve?

As we can view in the graph, the x-value falls closer to 50 than 45. Using the TI-83+, the invNorm command calculates the x-value from the amount of area shaded to the left, the population mean, and the standard deviation. Therefore the x-value is approximately 48.733.

Example G: What is the age at which 75% of more of the teachers at MHS are as old or older knowing the normal distribution of teacher ages has a mean of 40 years and standard deviation of 8 years.

The area shaded in the normal distribution represents the 75% of the teachers we could randomly select from so that their age would be as old or older than the cut-off x-value. That x-value falls between 32 and 40 as seen in the graph. Therefore, approximately 75% of the teachers are 34.6 years or older at MHS.

Random Variables and Normal CurvesWe have learned the difference between combining random variables and performing linear transformations on a random variable. We’ve also learned about normal probability distributions. Now we will put these concepts together.

In your own words describe the difference between combining random variables and performing linear transformations on a random variable and their effects on the resulting means and standard deviations.

Now apply the correct methods for combining random variables and performing linear transformations on random variables to the following problem involving random variables with normal probability distributions.

James and Laura run 2 miles a day to keep in shape for school athletics. James’ time is normally distributed with mean of 12 minutes and standard deviation of 1.5 minutes. Laura’s time is normally distributed with expected value of 13 minutes and variance

of 1 minute2. 1. Assuming James can maintain his pace, what is the expected run time if he doubles his distance to 4

miles? What is his new standard deviation?

2. What is the probability that James can run a 4-mile distance (assuming information in part a) in under 22 minutes?

3. What is Laura’s expect total run time for 5 days of 2-mile runs? What’s the standard deviation?

4. What is the probability the Laura’s total run time for 5 days of 2-mile runs will be at least 1 hour?

5. What is the expected difference between James and Laura’s 2-mile run time? What’s the standard deviation?

6. What is the probability that Laura’s run time will be less than James’ run time?

Review…Cell PhoneCell phone companies have different plans allowing customers to have varying number of minutes per plan. They know the total amount customers pay monthly for their basic service charge of plans, but is looking at gauging customers more. The chart represents the number of minutes customers go beyond their allotted time plan. For example, a plan allows for a total of 500 minutes per month but the customer used the phone for 560 minutes exceeding the plan by 60 minutes.

# of excess minutes, x 0 15 30 45 60 75 90 105

P(X = x) .37 .26 .17 .08 .05 .02 .02 .03

What is the expected number of excess minutes used? E(X) = ________

What is the variation of number of excess minutes? x2 = _________

To increase the cell phone’s profit, the company places a $.012 roaming charge on excess minutes beyond the plan allotment. What is the expected profit knowing the expected number of excess minutes? Also, what is the standard deviation of profit knowing the expected number of excess minutes? Let Y represent the profit based on excess minutes used on cell phones.

At Christmas, the cell phone gives all cell phone owners a holiday gift of 10 additional free minutes beyond their plan. What is the expected number of holiday excess minutes? What is the standard deviation of the holiday excess minutes? Let H represent the number of excess minutes used during the holiday month based on the expected number of excess minutes used regularly.

GradingThis is the grade distribution for the 2009 AP Statistics Exam at MHS.

AP Grade 1 2 3 4 5Probability 0.00 0.11 0.35 0.33 0.21

The Statistics Department decides to rescale the AP Grade to determine what grade would have been obtained if the test had been based on 100 points. The Statistics teacher works out a scaled grade based on Grade Scale = 60 + 6 . 868( AP Grade) . What is the scaled average and standard deviation of the scaled grade based on the expected AP Grade? The expected AP Grade is 3.64 and the variance is .8704.

Up, Up, and Away!A commuter airline flies small planes between Oklahoma City and Corpus Christi. For small planes, the baggage weight is a concern, especially on foggy mornings, because the weight of the plane has an effect on how quickly the plane can ascend. Suppose that it is known that the variable w = weight of baggage checked by a randomly selected passenger has a mean and standard deviation of 42 and 16 respectively. Consider a flight on which ten passengers, all traveling alone, are flying. If we use wi to denote the baggage weight for passenger i (for i ranging from 1 to 10), the total weight of checked baggage, t, is then t = w1 + w2 + …+ w10.

Note that t is a linear combination of the wi’s.

The expected value of t is __________.

Since the passengers are traveling alone, it is reasonable to think the ten baggage weights are unrelated and that the wi’s are independent. Then the variance of t is ____________.

Standardized Schmandardized!A nationwide standardized exam consists of a multiple-choice section and a free response section. For each section, the mean and standard deviation are reported to be

Mean Standard deviationMultiple-Choice 38 6Free Response 30 7

Let’s define X1 and X2 to be the multiple choice and free response score, respectively, of a student selected at random from those taking the exam. We are also interested in the variable Y = total score. Suppose that the total score is computed as Y = X1 + 2X2. What are the mean and standard deviation of Y?

Why can’t we calculate the standard deviation? What condition is required and how is it violated? __________________________________________________________________

SAT ScoresSuppose the mean SAT verbal score is 525 with standard deviation 100, while the mean SAT math score is 575 with standard deviation of 100. What can be said about the mean and standard deviation of the combined math and verbal scores? Calculate each if possible.