Chapter 8 (Inference)

Question 1

What are confidence intervals?

Answer 1

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

Question 2

What are hypothesis tests?

Answer 2

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

Question 3

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

Answer 3

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

Question 4

What is a confidence interval composed of?

Answer 4

A point estimate +- the margin or error

Question 5

What is the a point estimate?

Answer 5

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

Question 6

What is the margin of error?

Answer 6

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

Critical value * standard error

Question 7

What is a critical value?

Answer 7

t_{alpha/2}

Question 8

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

Answer 8

Alpha/2

Question 9

How to find a confidence interval for the mean

Answer 9

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

Question 10

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

Answer 10

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)

Question 11

What is the standard error of p_hat?

Answer 11

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

Question 12

What is the confidence level?

Answer 12

(1-alpha)100%

Question 13

Most common values for alpha?

Answer 13

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

If not given, assume alpha = .05

Question 14

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

Answer 14

To infer something meaningful about the population.

Question 15

The interval is _____

The parameter is ______

Answer 15

random,

fixed!!

Question 16

Which table do we use to infer the proportion?

Answer 16

The normal table!

Question 17

Which table do we use to infer the mean?

Answer 17

The t-table!

Question 18

A confidence interval makes a(n) ________,

A hypothesis test makes a(n) _______.

Answer 18

Estimate,

Comparison

Question 19

Hypothesis testing allows us to do what?

Answer 19

Asses the plausibility of a specific statement of hypothesis.

Question 20

The null and alternative hypothesis are…?

Answer 20

Complementary statements about the parameter of interests

Question 21

How to denote the null hypothesis?

Answer 21

H_o

h_naught

Question 22

How to denote the alternative hypothesis?

Answer 22

H_A

Question 23

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

Answer 23

Simple! Which means one value.

Question 24

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

Answer 24

Composite, the one with many values!

Question 25

What is a two sided hypothesis?

Answer 25

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

Question 26

What is a one sided hypothesis?

Answer 26

``` 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 ```

Question 27

What is a test statistic?

Answer 27

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

Question 28

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

Answer 28

``` T= sqrt(n)(X_bar - µ_o) ----–-–-–––––––––– s aka T= (X_bar - µ_o) ----–-–-–––––––––– s/sqrt(n) ```

Question 29

What is alpha?

Answer 29

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

Question 30

How do we denote the normal distribution of p_hat?

Answer 30

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

Question 31

What is the hypothesis test process?

Answer 31

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)

Question 32

What does the test statistic do?

Answer 32

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.

Question 33

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

Answer 33

against.

Question 34

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

Answer 34

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.

Question 35

What is a p-value?

Answer 35

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

Question 36

For H_A: µ ≠ µ_o, the p-value =

Answer 36

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

Question 37

For H_A: µ > µ_o, the p-value =

Answer 37

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

Question 38

For H_A: µ < µ_o, the p-value =

Answer 38

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

Question 39

What does a small p-value tell us?

Answer 39

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

Question 40

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

Answer 40

the significance level, or alpha, or size of the test. | Unless stated otherwise, we assume alpha = .05

Question 41

What is the significance level, or alpha?

Answer 41

The size of the test

Question 42

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

Answer 42

before the test is conducted.

Question 43

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

Answer 43

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

Question 44

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

Answer 44

We fail to reject the H_o.

Question 45

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

Answer 45

We may say that there is marginal evidence against the null

Question 46

Does statistical significance imply practical significance?

Answer 46

No.

Question 47

What does a hypothesis test determine?

Answer 47

If there is evidence against the null hypothesis

Question 48

What does a small p_value indicate?

Answer 48

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

Question 49

What is the confidence interval for proportion?

Answer 49

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

Question 50

What is the confidence interval for the mean?

Answer 50

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

Question 51

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

Answer 51

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