test 3

Question 1

what are zener cards

Answer 1

cards that test psychic powers, 5 card choices total

success/correct = choose 1 symbol and its correct meaning you predicted it right

Question 2

difference between N, n, and p

Answer 2

N: population size, total number of trials (ex. number of students)

n: number of experiments in which event of interest occurs (ex. number of cards)

p: probability of success

Question 3

what is a binomial distribution

Answer 3

used for determining probability of getting a certain number of successes, where each trial has only 2 possible outcomes: success or failure (used in zener decks)

Question 4

difference between normal and binomial distribution

Answer 4

binomial: used when you have a fixed number of trials (counting fails or successes)

normal: used for continuous data and can take on any interval, shaped like a bell-curve, no fixed number of trials

Question 5

explain how this sapply function works (which is used inside of data frames):

sapply(0:n, function(X) sum(x==X))

Answer 5

• sapply applies the given function
• 0:n generates a sequence from 0 to n number
• function(X) sum(x==X) is applied to each element of this sequence
• inside function, X represents each element of sequence
• for each X, it calculates how many times X occurs in the vector x (which was generated earlier)

The sum(x==X) part counts how many times the value X occurs in the vector x.

so in essence: sapply function is used to count how many times each number from 0 to n occurs in the generated x data

Question 6

what’s the code?

create a frequency distribution of the observed number correct in a zener deck with 25 cards and 100 students

Answer 6

first generate a set of data using the rbinom function

x <- rbinom(100, size=25, prob = 0.2)

prob of correct is 1/5 so prob. is 25, 100 students is N and size is 25 cards

now make a data frame for it:

df.zener <- data.frame(Count=0:25, Frequency = sapply(0:25, function(X) sum(x==X)))

this gives you the frequency distribution of the number correct

Question 7

mean and standard deviation formulas in a binomial distribution + codes for mean and mean prob.

Answer 7

mean = np

n = number of trials
p = prob. of success

standard deviation = √(np(1-p))

• these formulas always work for finding the mean and standard deviation in a binomial distribution

codes:
mean(x)
mean(x)/n for prob.
sd(x) for standard deviation

Question 8

95% confidence interval - why its used and how to code for it (using binomial distribution)

Answer 8

its used bc its reliable/precise and gives good info ab the data
- similar to 2 standard deviations above the mean

code:

qbinom(0.025, size= n, prob =p) #lower limit

qbinom(0.975, size=n, prob =p) #upper limit

q binom function is used for quartiles, and you are finding the 2.5th percentile and 97.5th percentile to get the 95% confidence interval

Question 9

what’s the code?

use the probability distribution function to get the probability of observing more than a certain value

Answer 9

pbinom(qbinom(0.975, size =n, prob=p), size =n , prob=p, lower.tail=FALSE)

pbinom for the probability, lower.tail=FALSE to find the upper threshold, the upper tail

Question 10

what’s the code?

what’s the probability that at least one person in N gets more than 9 cards correct?

Answer 10

prob. of no one getting more than 9 cards correct → (1-P)

then find the complement of that by subtracting it from 1 and raising to the N power → 1- (1-P)^N

code:
1 - (1 - pbinom(9, size=n, prob=p, lower.tail=FALSE))^N

Question 11

what’s the code?

finding the 95% confidence interval using the normal distribution

Answer 11

qnorm(0.025, mean= np, sd=sqrt(np*(1-p))) #lower limit

qnorm(0.975, mean= np, sd=sqrt(np*(1-p))) #upper limit

Question 12

formula to find Z score

Answer 12

Z = (X - μ)/σ

X = vector of observations
μ = mean
σ = standard deviation

Question 13

Z score (what it is, what values are common + Z score code)

Answer 13

Z score: tells u how far a particular data point is from the average of a group of data points, measured in terms of standard deviations

• Z score of 1 means 1 standard deviation and so on

Z 0.025 = -1.96 & Z 0.975 = 1.96

Z- score code: qnorm(0.975) or qnorm(0.025)

Question 14

what’s the code?

expected lower and upper 95% confidence limit of X using Z score

when it says expected, it has to do with proportions

Answer 14

qnorm(0.025) sqrt(np(1-p)) + np #lower limit

qnorm(0.975) sqrt(np(1-p)) + np #upper limit

formula: X =σZ + μ

Question 15

whats the code?

one sided z-test for proportions using p and p0 & two-sided z-test

Answer 15

z <- (p - p0) / sqrt(p0 * (1-p0)/N)

one sided
pnorm(z, lower.tail=TRUE)

• in one sided tests, we only care about the left hand side so (bc observed value is less than what we expected it to be)

what u just calculated gives u p value and if its less than 0.05, you reject null hypothesis, greater = accept null hypothesis

two sided
2*pnorm(abs(z), lower.tail=FALSE)

• can do either FALSE or TRUE for tail but just be consistent

Question 16

what are Q-Q plots used for?

Answer 16

to asses departures from normality

• whether or not data follows a theoretical distribution

Question 17

whats the code to find the empirical and expected (were the data normal) 99th percentile of weight

Answer 17

empirical means observed and you use the quantile function for that

for expected, you use the classic formula of X = σZp + μ

quantile(weight, prob=0.99) #empirical/observed 99th percentile

type=1 at the end is there by default

sd(weight)*qnorm(0.99)+mean(weight) #expected 99th percentile

what you’re calculating is basically the number that 99% of the data fall below

Question 18

how do you plot a Q-Q plot with the normal line too? (what’s the code for it)

use the weight example after attaching babies

Answer 18

qqnorm function creates the graph

qqnorm(weight, xlab=”theoretical quantiles”, ylab=”maternal weight (lbs)”) #q-q plot

qqline(weight, col=”red”) #normal line

Question 19

what is a t test used for

Answer 19

used to compare the means of 2 different populations

• more helpful when there are relatively small sample sizes and you are dealing with a normal distribution

Question 20

lower and upper limit values of the normal distribution

Answer 20

qnorm(0.025)
qnorm(0.975)

Z = -1.96 and 1.96

95% of the values on a standard normal distribution will be within this range