Flashcards in Chapter 7 - Hypothesis Testing (One-Sample) Deck (18):

1

## Hypothesis Testing

###
- framework for making decisions using probabilistic methods

- provides a uniform decision-making criterion

- one-sample = hypotheses are specified about a single distribution

- two-sample problem = two different distributions are compared

2

## Type I error

###
- probability of rejecting the null hypothesis when H0 is true

- it is referred to as the significance level of a test

- a test based on sample mean has the highest power among all tests with a given type I error

3

## Type II error

###
- probability of accepting the null hypothesis when H1 is true

- denoted by B

- function of the population mean

4

## Acceptance region

### - range of values for x for which H0 is not rejected (ie. it’s not that implausible)

5

## Rejection region

### - range of values for x for which H0 is rejected (ie. there’s enough evidence against H0)

6

## One-sided test

### - test in which the values of the parameter under the alternative hypothesis are allowed to be greater than or less than the values of the parameter under the null hypothesis, but not both

7

## Critical value method

### - critical value method = we compute a test statistic and determine the outcome of a test by comparing the test statistic with a critical value determined by the type I error

8

## p-value method

###
- the p-value is the significance level at which the given value of the test statistic is on the borderline between the acceptance and rejection regions

- can also be thought of as the probability of obtaining a test statistic as extreme as or more extreme than the actual test statistic obtained, given that the null hypothesis is true

9

## Statistical significance of a p-value

###
- if 0.01 < p < 0.05, then the results are significant

- if 0.001 < p < 0.1, then the results are highly significant

- if p < 0.001, then the results are very highly significant

- if p > 0.05, then the results are considered not statistically significant

- however, if 0.05 < p < 0.10, then a trend towards statistical significance is sometimes noted

10

## Statistical vs. scientific significance

###
- the results of a study can be statistically significant, but still can be not scientifically important

- eg. if a small difference was found to be statistically significant because of a large sample size

- some statistically non significant results can be scientifically important

- encourages researchers to perform larger studies

11

## Two-tailed test / Two-sided test

###
- the values of the parameter being studied under the alternative hypothesis are allowed to be either greater than or less than the values of the parameter under the null hypothesis

- the type I error is divided evenly between lower and upper rejection regions

12

## p-value for two-sided tests

### - because a two-sided alternative hypothesis is being used, extremeness is measures by the absolute value of the test statistic

13

## Decision between one-sided and two-sided

###
- it is easier to reject the null hypothesis using a one-sided test than using a two-sided test

- a two-sided test can be more conservative because it is not necessary to guess the appropriate side - - in some cases, only alternatives on one side of the null mean are of interests or are possible

- a one-sided test is better than a two-sided test because it has more power

- do not change from a two-sided to a one-sided test after looking at data

14

## Relationship between confidence intervals and two sided tests

###
- H0 is rejected with a two-sided test only if the two-sided CI for the mean does not contain the null mean

- H0 is accepted with a two-sided test only if the two-sided CI for the mean does contain the null mean

15

## Power of a test

###
- the calculation of power is used to plan a study, usually before any data have been obtained (exception = pilot study)

- can make a projection concerning the standard deviation without actually having any data to estimate it

- assume the standard deviation is known and base power calculations on the one-sample z test

16

## Factors affecting the power of a test

###
- if the significance level is made smaller, the power decreases

- if the alternative mean is shifter farther away from the null mean, then the power increases

- if the standard deviation of the distribution of individual observations increases, then the power decreases

- if the sample size increases, then the power increases

17

## Factors affecting the required sample size

###
- the sample size increases as the variance increases

- the sample size increases as the significance level decreases

- the sample size increases as the required power increases

- the sample size decreases as the absolute value of the distance between the null and alternative mean increases

18