OpenIntro 4

Question 1

Percentiles

Answer 1

Percentile is the percentage of observations that fall below a given data point. Graphically, percentile is the area below the probability distribution curve to the left of that observation.

Question 2

Z scores

Answer 2

Z = observation − mean / SD Z score of an observation is the number of standard deviations it falls above or below the mean. Z scores are defined for distributions of any shape, but only when the distribution is normal can we use Z scores to calculate percentiles. (Observations that are more than 2 SD away from the mean are usually considered unusual)

Question 3

Calculating the Z score

Answer 3

Z = observation − mean / SD

if z score = negative - it is below mean

if z score = positive - above the mean

Question 4

Pnorm() of Z score to get percentile

Answer 4

round(pnorm(1.50), digits=4)
output: 0.9332 = 93,32 % of possoms

Pnorm shows distribution up to the Z score

1 - pnorm shows the data from Z score and on

Question 5

Find the head lenght between 2 Z scores

Answer 5

subtract the 2 Z-scores from eachother

Question 6

Solving equation for percentile

Answer 6

Question 7

Calculating percentiles

Answer 7

There are many ways to compute percentiles/areas under the
curve:

In R:
> pnorm(1800, mean = 1500, sd = 300)
output: 0.8413447

Question 8

Heinz ketchup factory the amounts which go into bottles of ketchup
are supposed to be normally distributed with:

mean 36 oz.

standarddeviation 0.11 oz.

What percent of bottles have less than
35.8 ounces of ketchup?

Answer 8

pnorm(35.8, mean = 36, sd = 0.11)
output: 0.0345

Question 9

qnorm()

Answer 9

Question 10

Bernouilli random variable

Answer 10

only 2 outcomes

Question 11

Geometric distribution

If p represents probability of success, (1 − p) represents probability
of failure, and n represents number of independent trials
P(success on the nth trial) = (1 − p)n−1p

Answer 11

Geometric distribution needs:

1. independence: outcomes of trials don’t affect each other
2. identical: the probability of success is the same for each trial

Example from book:

On average, how many transistors would you expect to be produced before the first with a defect? What is the standard deviation?

1/0.02 = 50 made until one defect one

SD: sqrt (1 - 0.02 / 0.02^2) = 49.49

Question 12

roling a 6 (Geometric distribution)

Answer 12

Question 13

Suppose we randomly select four individuals to participate in this
experiment. What is the probability that exactly 1 of them will
refuse to administer the shock?

The Binomial distribution describes the probability of having
exactly k successes in n independent Bernouilli trials with
probability of success p.

Answer 13

Let’s call these people Allen (A), Brittany (B), Caroline (C), and
Damian (D). Each one of the four scenarios below will satisfy the
condition of “exactly 1 of them refuses to administer the shock”:

The probability of exactly one 1 of 4 people refusing to administer
the shock is the sum of all of these probabilities.
0.0961 + 0.0961 + 0.0961 + 0.0961 = 4 × 0.0961 = 0.3844

Question 14

When is it binomial distribution?

Answer 14

Question 15

What part does pnorm() return?

Answer 15

The stuff up to the input
Put 1 - to get the stuff after the input

Question 16

68-95-99.7 rule

Answer 16

Question 17

Further on Z scores

Answer 17

The z-score is positive if the value lies above the mean, and negative if it lies below the mean

A z-score describes the position of a raw score in terms of its distance from the mean, when measured in standard deviation units

Tells you “how far ahead, or how far below your raw score is from the population mean, in standard deviation units“.

Question 18

qnorm() and pnorm()

Answer 18

The qnorm() function is simply the inverse of pnorm()

fx:

> qnorm(.10) (percentile)
[1] -1.281552

> pnorm(-1.28) (Z-score)
[1] 0.1002726

Question 19

Find the cutoff

• the warmest 5 %

Answer 19

Question 20

The 4 conditions of the binomial distribution

Answer 20

1: The number of observations n is fixed.
2: Each observation is independent.
3: Each observation represents one of two outcomes (“success” or “failure”).
4: The probability of “success” p is the same for each outcome.

Question 21

The National Vaccine Information Center estimates that 90% of Americans have
had chickenpox by the time they reach adulthood.

(b) Calculate the probability that exactly 97 out of 100 randomly sampled American adults had chickenpox during childhood.
(c) What is the probability that exactly 3 out of a new sample of 100 American adults have not had chickenpox in their childhood?

Answer 21

> dbinom(97,100,.90)
[1] 0.005891602

> dbinom(03,100,.10)
[1] 0.005891602

Question 22

Geometric distribution - expected value

Answer 22

How many people is Dr. Smith expected to test before finding the first one that refuses to administer the shock?

The expected value, or the mean, of a geometric distribution is defined as

1 p . µ = 1 p = 1 0.35 = 2.86

She is expected to test 2.86 people before finding the first one that refuses to administer the shock.

Question 23

Choose function

Answer 23

k successes in n trials.

Note: You can also use R for these calculations: > choose(9,2)

output: 36

Question 24

A 2012 Gallup survey suggests that 26.2% of Americans are obese. Among a random sample of 10 Americans, what is the probability that exactly 8 are obese?

(binomial distribution)

Answer 24

(n=10 k=8) × (0.262)^8 × (0.738)^2 = 0.0005

Question 25

Expected value A 2012 Gallup survey suggests that 26.2% of Americans are obese. Among a random sample of 100 Americans, how many would you expect to be obese?

Answer 25

Easy enough, 100 × 0.262 = 26.2. Or more formally, µ = np = 100 × 0.262 = 26.2. But this doesn’t mean in every random sample of 100 people exactly 26.2 will be obese. In fact, that’s not even possible. In some samples this value will be less, and in others more. How much would we expect this value to vary?

Question 26

Distributions of number of successes

Answer 26

Low large is large enough? **The sample size is considered large enough if the expected number of successes and failures are both at least 10.** np ≥ 10 and n(1 − p) ≥ 10 (a) n = 100, p = 0.95 **(b) n = 25, p = 0.45 (correct answer)** (c) n = 150, p = 0.05 (d) n = 500, p = 0.015

Question 27

Normal approximation to the binomial (What is the probability that the average Facebook user with 245 friends has 70 or more friends who would be considered power users?)

Answer 27

Question 28

What is the probability that the average Facebook user with 245 friends has 70 or more friends who would be considered power users?

Answer 28

Question 29

Negative binomial distribution

Answer 29

The following four conditions are useful for identifying a negative binomial case: 1. The trials are independent. 2. Each trial outcome can be classified as a success or failure. 3. The probability of success (p) is the same for each trial. 4. The last trial must be a success. Note that the first three conditions are common to the binomial distribution.

Question 30

What is the probability that at most 3 out of 10 randomly sampled American adults have not had chickenpox? (binomial distribution)

Answer 30

Choose(10,0) \* 0.1^0\*0.9^10 + choose(10,1) \* 0.1^1\*0.9^9 + choose(10,2) \* 0.1^2\*0.9^8 + choose(10,3) \* 0.1^3\*0.9^7 [1] 0.9872048

Question 31

Binomial vs. negative binomial

Answer 31

Question 32

On 70% of days, a hospital admits at least one heart attack patient. On 30% of the days, no heart attack patients are admitted. Identify each case below as a binomial or negative binomial case, and compute the probability. (a) What is the probability the hospital will admit a heart attack patient on exactly three days this week? (b) What is the probability the second day with a heart attack patient will be the fourth day of the week? (c) What is the probability the fth day of next month will be the rst day with a heart attack patient?

Answer 32

In each part, p = 0.7. ## Footnote (a) The number of days is fixed, so this is binomial. The parameters are k = 3 and n = 7: 0.097. (b) The last "success" (admitting a heart attack patient) is fixed to the last day, so we should apply the negative binomial distribution. The parameters are k = 2, n = 4: 0.132. (c) This problem is negative binomial with k = 1 and n = 5: 0.006. Note that the negative binomial case when k = 1 is the same as using the geometric distribution.