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final exam Flashcards

1
Q

What is probability

A

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

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2
Q

What is a random experiment

A

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

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3
Q

What is the sample space

A

the set of all possible outcomes from a random experiment

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4
Q

What is an outcome

A

a result of a random experiment

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5
Q

What is an event of a random experiment

A

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

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6
Q

0 ≀ 𝑃 𝐸 ≀ 1

A

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

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7
Q

P(S) = 1

A

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

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8
Q

What properties must the sample space satisfy?

A
  • The outcomes in a sample space must be β€œexhaustive.”
  • The outcomes in a sample space must be β€œmutually exclusive.”
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9
Q

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

A

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

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10
Q

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

A

– No two outcomes can occur at the same time (on the same β€œtrial”).

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11
Q

P(E) = 1

A

π‘“π‘œπ‘Ÿ π‘Žπ‘›π‘¦ π‘ π‘’π‘žπ‘’π‘’π‘›π‘π‘’ π‘œπ‘“ π‘šπ‘’π‘™π‘‘π‘’π‘Žπ‘™π‘™π‘¦ 𝑒π‘₯𝑙𝑒𝑠𝑖𝑣𝑒 𝑒𝑣𝑒𝑛𝑑𝑠 𝐸1, 𝐸2, … , 𝐸𝑁, 𝑃 𝐸1 βˆͺ 𝐸2 … βˆͺ 𝐸𝑁 = the sum of 𝑃 𝐸𝑛

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12
Q

What does it mean for an experiment to be random?

A

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

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13
Q

Classical approach

A

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

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14
Q

Relative Frequency Approach

A

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

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15
Q

Subjective Approach

A

P(E) = your best guess

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16
Q

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

A

P(A) =/= P(B)

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17
Q

What is a random variable

A

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.

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18
Q

𝑃 (𝑋 = π‘₯𝑖) β‰₯ 0 𝑖𝑓 π‘₯𝑖 ∈ S

A

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

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19
Q

Sum of ∞ 𝑃( 𝑋 = π‘₯𝑖) = 1 𝑖𝑓 π‘₯𝑖 ∈ S

A

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

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20
Q

𝑃 (𝑋 = π‘₯) = 0 π‘“π‘œπ‘Ÿ π‘₯ βˆ‰ 𝑆

A

x values not included in the sample space can not occur

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21
Q

What is a test of independence?

A

P(A|B) = P(A)

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22
Q

What is the test for mutually exclusive?

A

P(A and B) = 0

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23
Q

What is a random variable?

A

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

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24
Q

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

A

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

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25
Q

is f(x) a probability?

A

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

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26
Q

What are the properties of a continuous random variable?

A

1) 𝑓 x β‰₯ 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 = π‘₯ = 0 π‘“π‘œπ‘Ÿ any x )
3) The total area under 𝑓 x is 1

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27
Q

What is the sampling distribution?

A

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

28
Q

What is the benefit of using a sampling distribution?

A

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.

29
Q

What is random sampling?

A

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.

30
Q

When can random sampling be called simple random sampling?

A

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

31
Q

What are the advantages of using SRS

A

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.

32
Q

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

A

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.

33
Q

What is the CLT?

A

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.

34
Q

What is the difference between point and interval estimators

A

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

35
Q

What is bias?

A

systematic error inherent in the estimate itself

36
Q

what is sampling error?

A

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

37
Q

When is the estimator unbiased?

A

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

38
Q

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

A

True

39
Q

What can make the CI wider?

A

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

40
Q

What is SE in a t-test?

A

Sx/ sqrt(𝑁)

41
Q

What is SE in a z-test?

A

Οƒ / sqrt(N)

42
Q

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

A

1.96(SE)

43
Q

How do you find the CI?

A

(X - MOE, X + MOE)

44
Q

How do you find Z?

A

X - u = 𝜎/ sqrt(N)

45
Q

What is the a level?

A

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.

46
Q

What should you include in the statistical test?

A

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)

47
Q

What is Null Hypothesis Significance Testing?

A

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

48
Q

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

A

when z is positive

49
Q

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

A

when Ha is nondirectional(two-tailed)

50
Q

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

A

Either Ho is false or Ho is true

51
Q

What is the p value

A

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.

52
Q

When do you use a/2?

A

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

53
Q

The critical z value(s) for a 1-sample z test when the alternative hypothesis is that ΞΌ > ΞΌ0 and the significance level is .01.

A

qnorm(.01, lower.tail = FALSE)

54
Q

The critical z value(s) for a 1-sample z test when the alternative hypothesis is that ΞΌ β‰  ΞΌ0 and the significance level is .01.

A

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

55
Q

The p-value for a 1-sample z test when the alternative hypothesis is that ΞΌ > ΞΌ0 and the observed z score is 1.40.

A

> pnorm(1.40, lower.tail = FALSE)

56
Q

The p-value for a 1-sample z test when the alternative hypothesis is that ΞΌ β‰  ΞΌ0 and the observed z score is –2.20.

A

2 * pnorm(-2.20)

57
Q

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

A

p * 2

58
Q

The critical t value(s) for a 1-sample t test when the alternative hypothesis is that ΞΌ < ΞΌ0, the sample size is 52, and the significance level is .05. You may use r

A

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

59
Q

The critical t value(s) for a 1-sample t test when the alternative hypothesis is that ΞΌ β‰  ΞΌ0, the sample size is 80, and Ξ± = .01.

A

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

60
Q

The p-value for a 1-sample t test when the alternative hypothesis is that ΞΌ < ΞΌ0, the sample size is 39, and the observed t score is –2.55.

A

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

61
Q

The p-value for a 1-sample t test when the alternative hypothesis is that ΞΌ β‰  ΞΌ0, the sample size is 75, and the observed t score is 1.82.

A

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

62
Q

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

A

[X - MOE , X + MOE]

63
Q

What is the MOE for a t- test?

A

2(SE)

64
Q

What is SE for t?

A

Sx/ sqrt(N)

65
Q

How do you find t?

A

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

66
Q

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

A

pt(t, df = N-1)

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