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Stat 3000 > Chapter 8 (Inference) > Flashcards

Chapter 8 (Inference) Flashcards

1
Q

What are confidence intervals?

A

They facilite the estimation of population parameters, address questions like “what is mean body temperature?”

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2
Q

What are hypothesis tests?

A

They enable us to compare a population parameter to a specified value. Address questions such as “is the mean body temperature 98.6?”

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3
Q

What is a (1-alpha)100% confidence interval? (CI)

A

For an unknown parameter it Contains a set of plausible values of the parameter that are consistent with the data.

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4
Q

What is a confidence interval composed of?

A

A point estimate +- the margin or error

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5
Q

What is the a point estimate?

A

The value of the statistic. Asked in the chosen sample.

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6
Q

What is the margin of error?

A

It’s based on the standard error and a desired level of confidence.

Critical value * standard error

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7
Q

What is a critical value?

A

t_{alpha/2}

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8
Q

What is the area to the right of a t_{alpha/2}?

A

Alpha/2

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9
Q

How to find a confidence interval for the mean

A

X_bar ± (t_{alpha/2, n-1}) * s/sqrt(n)

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10
Q

What is the difference between the standard deviation and the standard error?

A

The standard deviation is how much a data set varies
s.d. = sqrt(sigma^2)

The standard error is how much our statistic distribution set varies
s.e. = sigma/sqrt(n)

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11
Q

What is the standard error of p_hat?

A

s.e.(p_hat) = sqrt((p_hat(1-p_hat)/n)

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12
Q

What is the confidence level?

A

(1-alpha)100%

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13
Q

Most common values for alpha?

A

.05, or .01.
Thus CIs are often 95% or 99%

If not given, assume alpha = .05

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14
Q

Why do we accept some probability that the interval doesn’t cover our parameter?

A

To infer something meaningful about the population.

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15
Q

The interval is _____

The parameter is ______

A

random,

fixed!!

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16
Q

Which table do we use to infer the proportion?

A

The normal table!

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17
Q

Which table do we use to infer the mean?

A

The t-table!

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18
Q

A confidence interval makes a(n) ________,

A hypothesis test makes a(n) _______.

A

Estimate,

Comparison

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19
Q

Hypothesis testing allows us to do what?

A

Asses the plausibility of a specific statement of hypothesis.

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20
Q

The null and alternative hypothesis are…?

A

Complementary statements about the parameter of interests

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21
Q

How to denote the null hypothesis?

A

H_o

h_naught

22
Q

How to denote the alternative hypothesis?

A

H_A

23
Q

The H_o takes on the _______ statement/portion of the question.

A

Simple! Which means one value.

24
Q

The H_A takes on the _______ statement/portion of the question.

A

Composite, the one with many values!

25
Q

What is a two sided hypothesis?

A

Where H_A is simply not H_o
H_o: µ = µ_o
H_A: µ ≠ µ_o

26
Q

What is a one sided hypothesis?

A 
Where H_A > H_o
-  H_o: µ = µ_o
versus H_A: µ > µ_o
or
Where H_A < H_o
- H_o: µ = µ_o
versus H_A: µ < µ_o
27
Q

What is a test statistic?

A

A numeric summary of our experimental results that can be used to determine how consistent the data are with the null hypothesis.

28
Q

When conducting a hypothesis test for a MEAN, what test statistic will we use?

A 
T= sqrt(n)(X_bar - µ_o)
----–-–-––––––––––
s
aka
T= (X_bar - µ_o)
----–-–-––––––––––
s/sqrt(n)
29
Q

What is alpha?

A

The probability that the confidence interval does NOT cover the parameter.

30
Q

How do we denote the normal distribution of p_hat?

A

p_hat ~ N (p, (p(1-p))/n)

31
Q

What is the hypothesis test process?

A
  1. State hypotheses about the parameter
  2. Collect data
  3. Construct a test statistic
  4. Compute a p-value
  5. Draw conclusions (in statistical terms and in context)
32
Q

What does the test statistic do?

A

It quantifies how different what we observe is from what we expect.
It takes into account how much we’d expect the value of the statistic to vary by chance.

33
Q

When we observe an extreme value for our test statistic, we have evidence ________ the null hypothesis

A

against.

34
Q

When conducting a hypothesis test for a PROPORTION, what test statistic will we use?

A

Z = (p_hat - p_o) / sqrt[(p_o *(1-p_o))/n] ~ N(01)

using p_o to compute the s.e. since, under H-o, we assume p_o is the true value of p.

35
Q

What is a p-value?

A

the probability of obtaining the data we observed or data more extreme (less consistent with H-o) IF the null hypothesis is true.

36
Q

For H_A: µ ≠ µ_o, the p-value =

A

P(|T| > t_{n-1} | H_o is true)

37
Q

For H_A: µ > µ_o, the p-value =

A

P(T > t_{n-1} | H_o is true)

38
Q

For H_A: µ < µ_o, the p-value =

A

P(T < t_{n-1} | H_o is true)

39
Q

What does a small p-value tell us?

A

That our data are unlikely to occur if the null hypothesis is true, thus it provides evidence agains H_o.

40
Q

In the term “small” p-value, what is small defined by?

A

the significance level, or alpha, or size of the test.

Unless stated otherwise, we assume alpha = .05

41
Q

What is the significance level, or alpha?

A

The size of the test

42
Q

When should the significance level, or alpha, be specified?

A

before the test is conducted.

43
Q

If p < alpha, do we accept/reject the H_o?

A

We reject H_o, or reject the null, and say the results are statistically significant.

44
Q

If p ≥ alpha, do we accept/reject the H_o?

A

We fail to reject the H_o.

45
Q

If p > alpha but close to it, do we accept/reject the H_o?

A

We may say that there is marginal evidence against the null

46
Q

Does statistical significance imply practical significance?

A

No.

47
Q

What does a hypothesis test determine?

A

If there is evidence against the null hypothesis

48
Q

What does a small p_value indicate?

A

That the null hypothesis isn’t plausible, and thus there is evidence against the null hypothesis.

49
Q

What is the confidence interval for proportion?

A

(p_hat ± (Z_{alpha/2} * sqrt[p_hat(1-p_hat)/n])

50
Q

What is the confidence interval for the mean?

A

(X_bar ± ( t_{alpha/2,n-1} * (s/sqrt[n]))

51
Q

What is the connection between confidence intervals and the hypotheses test?

A

The confidence interval values are the same values for which we would fail to reject the null hypothesis.

Stat 3000 (12 decks)

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