final exam

Question 1

What is probability

Answer 1

a mathematical function of an event in a sample space, quantifying the likelihood of that event occurring in accordance with specific axiomatic rules.

Question 2

What is a random experiment

Answer 2

a well-defined procedure or action that produces an (observable) outcome in the sample space.

Question 3

What is the sample space

Answer 3

the set of all possible outcomes from a random experiment

Question 4

What is an outcome

Answer 4

a result of a random experiment

Question 5

What is an event of a random experiment

Answer 5

A subset of the sample space or a set of outcomes in the sample space.

Question 6

0 ≤ 𝑃 𝐸 ≤ 1

Answer 6

Probability of any event must lie between 0 and 1, inclusive

Question 7

P(S) = 1

Answer 7

Probability that any of the outcomes in S occurs must be 1.

Question 8

What properties must the sample space satisfy?

Answer 8

• The outcomes in a sample space must be “exhaustive.”
• The outcomes in a sample space must be “mutually exclusive.”

Question 9

What does it mean for outcomes in S to be exhaustive?

Answer 9

– All possible outcomes must be listed/
– Each “trial” (or experiment) must result in one of these outcomes.

Question 10

What does it mean for outcomes in S to be mutually exclusive?

Answer 10

– No two outcomes can occur at the same time (on the same “trial”).

Question 11

P(E) = 1

Answer 11

𝑓𝑜𝑟 𝑎𝑛𝑦 𝑠𝑒𝑞𝑢𝑒𝑛𝑐𝑒 𝑜𝑓 𝑚𝑢𝑙𝑡𝑢𝑎𝑙𝑙𝑦 𝑒𝑥𝑙𝑢𝑠𝑖𝑣𝑒 𝑒𝑣𝑒𝑛𝑡𝑠 𝐸1, 𝐸2, … , 𝐸𝑁, 𝑃 𝐸1 ∪ 𝐸2 … ∪ 𝐸𝑁 = the sum of 𝑃 𝐸𝑛

Question 12

What does it mean for an experiment to be random?

Answer 12

An experiment whose outcome cannot be predicted, but the possible outcomes can be listed

Question 13

Classical approach

Answer 13

P(E) = Number of possible outcomes in which E occurs
/ Total number of possible outcomes
(Assume outcomes are equally likely (flipping coins))

Question 14

Relative Frequency Approach

Answer 14

P(E)= Number of trials in which E occurs/ Total number of trials.
(assign probabilities on the basis of data)

Question 15

Subjective Approach

Answer 15

P(E) = your best guess

Question 16

What does P(A|B) =/= P(B|A) mean?

Answer 16

P(A) =/= P(B)

Question 17

What is a random variable

Answer 17

A random variable (X) is a real-valued function of an event or a set of
outcomes of a random experiment to a numerical value.

Question 18

𝑃 (𝑋 = 𝑥𝑖) ≥ 0 𝑖𝑓 𝑥𝑖 ∈ S

Answer 18

For all x values/ outcomes in the sample space, the probabilities must be positive.

Question 19

Sum of ∞ 𝑃( 𝑋 = 𝑥𝑖) = 1 𝑖𝑓 𝑥𝑖 ∈ S

Answer 19

The sum of all probabilities of the possible x values in the sample space must be 1

Question 20

𝑃 (𝑋 = 𝑥) = 0 𝑓𝑜𝑟 𝑥 ∉ 𝑆

Answer 20

x values not included in the sample space can not occur

Question 21

What is a test of independence?

Answer 21

P(A|B) = P(A)

Question 22

What is the test for mutually exclusive?

Answer 22

P(A and B) = 0

Question 23

What is a random variable?

Answer 23

a real valued function of an event or a set of outcomes of a random experiment

Question 24

What’s the difference between a continuous and discrete random variable.

Answer 24

Continuous: can take on any real value within an interval.
Discrete: can take on a countable number of possible values.

Question 25

is f(x) a probability?

Answer 25

no, but the area under f(x) can be interpreted as one.

Question 26

What are the properties of a continuous random variable?

Answer 26

1) 𝑓 x ≥ 0 2) The area under 𝑓 x can be interpreted as a probability - e.g.,) P(100 < X < 120) - 𝑓 x itself is not a probability (𝑃 𝑋 = 𝑥 = 0 𝑓𝑜𝑟 any x ) 3) The total area under 𝑓 x is 1

Question 27

What is the sampling distribution?

Answer 27

the probability distribution of a sample statistic for a given sample size (N)

Question 28

What is the benefit of using a sampling distribution?

Answer 28

allows us to make statistical inferences about population parameters using a sample. – to quantify the uncertainty or margin of error of our estimates. – to form the basis for conducting statistical tests on population parameters.

Question 29

What is random sampling?

Answer 29

a process of selecting a subset of cases from a larger population at random. – Each case in the population has a known probability of being selected to be part of the sample.

Question 30

When can random sampling be called simple random sampling?

Answer 30

In a simple random sampling or SRS, each case has an equal chance of being selected.

Question 31

What are the advantages of using SRS

Answer 31

1) It increases the likelihood that the sample is representative of the population. – What if the sample size is extremely large? 2) It allows us to establish the probability distribution of a sample statistic.

Question 32

Why can we treat a sample statistic as a random variable in random sampling?

Answer 32

The data from the sample go through a function to output a numerical value of the sample statistic, and depending on the sample the value will be different, so it will have a probability distribution.

Question 33

What is the CLT?

Answer 33

When a sample is collected through random sampling, and N is sufficiently large, x approximately follows normal distribution regardless of the population mean. X will get closer and closer to mu.

Question 34

What is the difference between point and interval estimators

Answer 34

A point estimator estimates a single value of a population parameter and an interval estimator estimates a range of values where the true population parameter could fall in certain probabilities

Question 35

What is bias?

Answer 35

systematic error inherent in the estimate itself

Question 36

what is sampling error?

Answer 36

random errors occurring as the parameter is based on a sample. Unavoidable when not evaluating the population as a whole.

Question 37

When is the estimator unbiased?

Answer 37

If the expected value of a parameter is equal to the true value of the parameter

Question 38

If X is normal, then estimated x follows normal even if the sample is not equal or greater than 30

Answer 38

True

Question 39

What can make the CI wider?

Answer 39

reducing a, increasing confidence level, reducing N, when pop sd is larger

Question 40

What is SE in a t-test?

Answer 40

Sx/ sqrt(𝑁)

Question 41

What is SE in a z-test?

Answer 41

σ / sqrt(N)

Question 42

What is MOE for a 95% CI in a Z- Test?

Answer 42

1.96(SE)

Question 43

How do you find the CI?

Answer 43

(X - MOE, X + MOE)

Question 44

How do you find Z?

Answer 44

X - u = 𝜎/ sqrt(N)

Question 45

What is the a level?

Answer 45

the chosen threshold for saying that the probability of a test-statistic at least as extreme as the observed one is small enough to reject H0.

Question 46

What should you include in the statistical test?

Answer 46

1) type of test, 2) research hypothesis, 3)description of procedure w/ summary stats (sample mean, pop mean, pop sd), 4) reason for decision, 5)statistical conclusion, 6) conclusion/ implication, 6) relevant stats (z/t & p)

Question 47

What is Null Hypothesis Significance Testing?

Answer 47

a statistical procedure that determines whether there is enough evidence to support a research hypothesis about population parameter(s)

Question 48

When do you use pnorm(z, lower.tail = FALSE)

Answer 48

when z is positive

Question 49

when do you use 2 * pnorm(z, lower.tail = FALSE)

Answer 49

when Ha is nondirectional(two-tailed)

Question 50

What does it mean that Ho and Ha must be mutually exclusive and exhaustive?

Answer 50

Either Ho is false or Ho is true

Question 51

What is the p value

Answer 51

The p-value is the chance of observing a result (e.g., a mean) as extreme as, or more extreme than, your result (e.g., your value of 𝑋), assuming that H0 is true. – A low p-value is evidence against H0, but it is not the probability of H0.

Question 52

When do you use a/2?

Answer 52

When you are finding the z value or t value w/ qnorm or qt and the test is two tailed

Question 53

The critical z value(s) for a 1-sample z test when the alternative hypothesis is that μ > μ0 and the significance level is .01.

Answer 53

qnorm(.01, lower.tail = FALSE)

Question 54

The critical z value(s) for a 1-sample z test when the alternative hypothesis is that μ ≠ μ0 and the significance level is .01.

Answer 54

> qnorm(0.005) [1] -2.575829 > qnorm(0.005, lower.tail = FALSE) [1] 2.575829

Question 55

The p-value for a 1-sample z test when the alternative hypothesis is that μ > μ0 and the observed z score is 1.40.

Answer 55

> pnorm(1.40, lower.tail = FALSE)

Question 56

The p-value for a 1-sample z test when the alternative hypothesis is that μ ≠ μ0 and the observed z score is –2.20.

Answer 56

2 * pnorm(-2.20)

Question 57

How do you find p for a two tailed test if you already have p for a one tailed test?

Answer 57

p * 2

Question 58

The critical t value(s) for a 1-sample t test when the alternative hypothesis is that μ < μ0, the sample size is 52, and the significance level is .05. You may use r

Answer 58

round(qt(.05, df = 51), 3)

Question 59

The critical t value(s) for a 1-sample t test when the alternative hypothesis is that μ ≠ μ0, the sample size is 80, and α = .01.

Answer 59

round(qt(.005, df = 79), 3)

Question 60

The p-value for a 1-sample t test when the alternative hypothesis is that μ < μ0, the sample size is 39, and the observed t score is –2.55.

Answer 60

round(pt(-2.55, df = 38), 3)

Question 61

The p-value for a 1-sample t test when the alternative hypothesis is that μ ≠ μ0, the sample size is 75, and the observed t score is 1.82.

Answer 61

pt(1.82, df = 74, lower.tail = FALSE) *2

Question 62

How to you find a 95% CI for a t-test?

Answer 62

[X - MOE , X + MOE]

Question 63

What is the MOE for a t- test?

Answer 63

2(SE)

Question 64

What is SE for t?

Answer 64

Sx/ sqrt(N)

Question 65

How do you find t?

Answer 65

X - u / (Sx/ sqrt(N))

Question 66

r formula for finding p for on tailed t- test:

Answer 66

pt(t, df = N-1)