unit 2 - chapter 9 - hypothesis testing with 1 sample

Question 1

6 steps of a hypothesis test

Answer 1

1. State the hypotheses
2. Set alpha
3. Calculate the test statistic
4. Determine the critical value
5. Make a test decision
6. Apply test result to context

Question 2

4 facets of the null (Ho)

Answer 2

1 - the status quo
2 - everything is unrelated
3 - no difference between groups
4 - everything arises from chance

part of step 1 - state the hypotheses

Question 3

the alternative

Answer 3

(H1/HA)
The opposite of the null (Ho)

Question 4

Ho and H1 are

Answer 4

Like the sun and the moon… Ho and H1 are mutually exclusive and collectively exhaustive (cover all possible outcomes)

Question 5

status quo meaning

Answer 5

The current baseline (or status quo) is taken as a reference point, and any change from that baseline is perceived as a loss or gain. Corresponding to different alternatives, this current baseline or default option is perceived and evaluated by individuals as a positive.

Question 6

step 2 - select alpha

Answer 6

Alpha →
Level of significance
Statistically significant
Our risk tolerance

Common alphas
0.10, 0.05, 0.01, 0.01

Question 7

step 3 - calculate the test stat

Answer 7

1 - take a sample
2 - compute the summary statistics
3 - run the appropriate test

Question 8

step 4 - determine the critical value

Answer 8

1 - alpha
2 - 1 or 2 tail test
3 - type of test

Question 9

important step 5 - make your test decision

Answer 9

TS > CV –> reject the null
reject when test stat is large
TS < CV –> fail to reject the null
fail when test stat is small
small test stat is a fail

Question 10

Failing to reject Ho does not mean Ho is true

Answer 10

true

Question 11

type 1 vs type 2 error

Answer 11

Type 1 error
Rejecting a true null
Probability is alpha (risk of making a type 1 error)

Type 2 error
Failing to reject a false null
Probability is beta

Question 12

Whether the null is true or false lies outside any test we do

Answer 12

true

We may not know where our population is
Alpha is static (will stay the same)
Not 1 - alpha

Question 13

step 6 - apple rest result to context (conceptual)

Answer 13

Context: if i switch to ameriprise auto insurance I can save $532 or more
Ho mew <= $532
H1 mew > $532

Question 14

test decision - 2x2 hypothesis test decision matrix

Answer 14

Ho is true + ftr Ho
correct decision

Ho is true + reject Ho
incorrect (type 1 - a)

Ho is false + ftr Ho
incorrect (type 2 - b)

Ho is false + reject Ho
correct decision

when you reject = new action taken
when you ftr = no new action

Question 15

Ho is true + ftr Ho

Answer 15

correct decision
free innocent

Question 16

Ho is true + reject Ho

Answer 16

incorrect (type 1 - a)
convict innocent

Question 17

Ho is false + ftr Ho

Answer 17

incorrect (type 2 - b)
free guilty

Question 18

Ho is false + reject Ho

Answer 18

correct decision
convict guilty

Question 19

alpha in tests

Answer 19

Certain beyond a reasonable doubt

Question 20

z tests - assumptions for 1 sample test

Answer 20

1 - the null is true –> always testing the null
2 - random sampling
3 - central limit theorem is satisfied
4 - interval or ratio data

Question 21

the test stat equation

Answer 21

TS = (xbar - mew) / sxbar

xbar = random variable
mew = population mean
sxbar = standard error of the mean

TS = observed value - expected value/chance

Question 22

what is SEM and why do we need it

Answer 22

Chance is randomness/error → SEM

we need to have a measure of relative distance

SEM = s/square root of n
standard deviatio adjusted by sample size

Question 23

normal distribution: converting z to Z score

Answer 23

Z = (x-x bar)/s

z = random variable
x bar = sample mean
s = standard deviation

Question 24

1 sample test for the mean: converting x bar to test stat

Answer 24

TS = xbar - mew/ sxbar

xbar = random variable
mew = population mean
sxbar = standard error of the mean

Question 25

important:

if we reject then we apply…
if we fail to reject then we apply…

Answer 25

reject = apply alternative back to context of problem. there is statistical significance

fail to reject = apply hypothesis back to context of the problem. statistically similar.

Question 26

important: why is the alpha critical line so far out on the tail

Answer 26

There’s a reason for why it’s so far out (loose numbers is in the center)
Possible type 1 error: rejecting a true null (alpha)
The result is not due to a real reason but due to chance

low score = chance so 1 data point =/= trend

Question 27

t vs z distibutions

Answer 27

• Z and t distributions are both symmetric; centered at mew = 0
• t distributions shape approaches Z at n = 30
• t distributions wider tails at n < 30
• t distributions account for the greater uncertainty (increase in variation) associated with smaller samples

Question 28

lenon pregnancy and hypothesis

Answer 28

Ho = lennon is without child
H1 = lennon is pregnant

Question 29

confidence intervals

Answer 29

One job of a statistician is to make statistical inferences about populations based on samples taken from the population. Confidence intervals are one way to estimate a population parameter. Another way to make a statistical inference is to make a decision about the value of a specific parameter.

Question 30

hypothesis testing

Answer 30

A statistician will make a decision about these claims. This process is called “hypothesis testing.” A hypothesis test involves collecting data from a sample and evaluating the data. Then, the statistician makes a decision as to whether or not there is sufficient evidence, based upon analyses of the data, to reject the null hypothesis.

Question 31

null hypothesis

Answer 31

H0: The null hypothesis: It is a statement of no difference between the variables–they are not related. This can often be considered the status quo and as a result if you cannot accept the null it requires some action.

Question 32

alternative hypothesis

Answer 32

Ha: The alternative hypothesis: It is a claim about the population that is contradictory to H0 and what we conclude when we cannot accept H0. This is usually what the researcher is trying to prove. The alternative hypothesis is the contender and must win with significant evidence to overthrow the status quo.

This concept is sometimes referred to the tyranny of the status quo because as we will see later, to overthrow the null hypothesis takes usually 90 or greater confidence that this is the proper decision.

Question 33

There are two options for a decision. They are “cannot accept H0” if the sample information favors the alternative hypothesis or “do not reject H0” or “decline to reject H0” if the sample information is insufficient to reject the null hypothesis.

Answer 33

Ho

> =
<=

H1
=/=
<
>

Question 34

type 1 and type 2 error from book

Answer 34

α = probability of a Type I error = P(Type I error) = probability of rejecting the null hypothesis when the null hypothesis is true: rejecting a good null.

β = probability of a Type II error = P(Type II error) = probability of not rejecting the null hypothesis when the null hypothesis is false. (1 − β) is called the Power of the Test.

α and β should be as small as possible because they are probabilities of errors.

Question 35

critical values

Answer 35

z = (x-xbar)/s

critical values, are marked on the bottom panel as the Z values associated with the probability the analyst has set as the level of significance in the test, (α). The probabilities in the tails of both panels are, therefore, the same.
Notice that for each 𝑋⎯⎯⎯there is an associated Zc, called the calculated Z, that comes from solving the equation above.

This calculated Z is nothing more than the number of standard deviations that the hypothesized mean is from the sample mean. If the sample mean falls “too many” standard deviations from the hypothesized mean we conclude that the sample mean could not have come from the distribution with the hypothesized mean, given our pre-set required level of significance.

Question 36

p value approach

Answer 36

Simply stated, the p-value approach compares the desired significance level, α, to the p-value which is the probability of drawing a sample mean further from the hypothesized value than the actual sample mean. A large p-value calculated from the data indicates that we should not reject the null hypothesis.

The smaller the p-value, the more unlikely the outcome, and the stronger the evidence is against the null hypothesis. We would reject the null hypothesis if the evidence is strongly against it.

Question 37

alpha vs p value

Answer 37

If α > p-value, cannot accept H0. The results of the sample data are significant. There is sufficient evidence to conclude that H0 is an incorrect belief and that the alternative hypothesis, Ha, may be correct.

If α ≤ p-value, cannot reject H0. The results of the sample data are not significant. There is not sufficient evidence to conclude that the alternative hypothesis, Ha, may be correct. In this case the status quo stands.

When you “cannot reject H0”, it does not mean that you should believe that H0 is true. It simply means that the sample data have failed to provide sufficient evidence to cast serious doubt about the truthfulness of H0.

Question 38

one-tailed tests

Answer 38

It may be that the analyst has no concern about the value being “too” high or “too” low from the hypothesized value. If this is the case, it becomes a one-tailed test and all of the alpha probability is placed in just one tail and not split into α/2 as in the above case of a two-tailed test. Any test of a claim will be a one-tailed test.
For example, a car manufacturer claims that their Model 17B provides gas mileage of greater than 25 miles per gallon. The null and alternative hypothesis would be:
H0: µ ≤ 25
Ha: µ > 25

Question 39

effects of sample size on test statistics

Answer 39

In developing the confidence intervals for the mean from a sample, we found that most often we would not have the population standard deviation, σ. If the sample size were less than 30, we could simply substitute the point estimate for σ, the sample standard deviation, s, and use the student’s t-distribution to correct for this lack of information.

When testing hypotheses we are faced with this same problem and the solution is exactly the same. Namely: If the population standard deviation is unknown, and the sample size is less than 30, substitute s, the point estimate for the population standard deviation, σ, in the formula for the test statistic and use the student’s t-distribution.

If we do not know σ, but the sample size is 30 or more, we simply substitute s for σ and use the normal distribution.

Question 40

one tailed test vs two tailed test

Answer 40

hypothesized values of p (probability)…

two tailed=
Ho: p = po
H1: p =/= po

one tailed=

Ho: p<=po
H1: p>po
either direction