TEST 2 REVIEW

population mean (µ) hypothesis test: - t values ( n ≤ 50) are found in

body of table based on α and degrees of freedom (table 4)

critical values are found

body or bottom (when sufficiently large) of t-table

population mean (µ) hypothesis test: - z scores (n ≥ 50) are found

bottom of table (table 4)

is a claim regarding the value of a single or multiple population characteristics

hypothesis

lower tail

left tailed

α is called

level of significance

right-tailed test: - t-test

look up value using α and df and keep POSITIVE

left-tailed test: - t-test

look up value using α and df and make value NEGATIVE

sample statistic will be in the _____ of the confidence interval

middle

the t distribution is very similar to the normal distribution but has more?

more variability than the normal distribution (z-distribution)

lower the level of confidence

narrower the interval

use ____ to create an interval _________. this interval contains more values and we can be a certain levels of confident it contains the _______.

confidence intervals around the sample true population parameter

the proportion of intervals which would contain the population parameter if we conducted a large number of samples

confidence level

the total amount of area will be the same for?

the same α

the added variability in the t-distribution is due to

the sample size - also know as the degrees of freedom (df: n-1)

p + q =

1

α= .01

1% represent rejection region 99% represents fail-to-reject region

using the t-table

1. find df (n-1) in first or last column 2. determine if it's one tail (α) or two tailed (α/2) 3. use α or α/2 and find the intersection of the df and tα to determine the critical value - z critical values are at the bottom of t-table

q̂ equals

1-p̂

5 steps to conduct hypothesis test

1. determine null and alternative hypothesis 2. calculate test statistic 3. find p-value OR critical value 4. make decision 5. state conclusion

how can inference be made from (n ≤ 51 or n < 30) small samples?

1. n is randomly selected from the population of interest 2. the population of interest is approximately normally distributed - we use t distribution

2 types of hypothesis are

1. null hypothesis- Ho: no difference 2. alternative hypothesis- Ha: difference

Sampling distribution for x̄: -requirements for large samples (z-distributions)

1. random sample form population of interest 2. sample size is sufficiently large (n ≥ 51)

Sampling distribution for x̄: -requirements for small samples (t-distributions)

1. random sample from population of interest 2. population of interest is approximately normal (use table 4)

α= .05

5% represents rejection region 95% represents fail-to-reject region

left-tailed test

Ha: parameter < "value" - p < p₀ - µ < µ₀

right-tailed test

Ha: parameter > "value" - p > p₀ - µ > µ₀

two-tailed test

Ha: parameter ≠ "value" - p ≠ p₀ - µ ≠ µ₀

Population proportion (p) hypothesis test: - left tailed

Ho (null): p = p₀ Ha (alternative): p < p₀ - population proportion is LESS THAN status quo suggests

Population proportion (p) hypothesis test: - right tailed

Ho (null): p = p₀ Ha (alternative): p > p₀ - population proportion is GREATER TAHN status quo suggests

Population proportion (p) hypothesis test: - two tailed

Ho (null): p = p₀ Ha (alternative): p ≠ p₀ - population proportion IS NOT EQUAL to status quo

Population mean (µ) hypothesis test: - right tailed

Ho (null): µ = µ₀ Ha (alternative): µ > µ₀ - population mean is GREATER THAN status quo suggests

Population mean (µ) hypothesis test: - two tailed

Ho (null): µ = µ₀ Ha (alternative): µ ≠ µ₀ - population mean IS NOT EQUAL to status quo

Population mean (µ) hypothesis test: - left tailed

Ho (null): µ = µ₀ Ha (alternative): µ < µ₀ - population mean is LESS THAN status quo suggest

referred to as the researchers hypothesis, is the claim that competes with the null hypothesis. usually the claim researchers try to support with evidence

alternative hypothesis - researcher wants to reject the null hypothesis

fail to reject null hypothesis (Ho)

at α ("value") level of significance, there IS NOT SUFFICIENT EVIDENCE to suggest that the alternative hypothesis is accurate. - replace α and Ha based on problem

reject null hypothesis (Ho)

at α ("value") level of significance, there IS SUFFICIENT EVIDENCE to suggest that the alternative hypothesis is accurate. - replace α and Ha based on problem

tells us how to estimate the population parameter from the sample data

confidence interval

values which represent extreme values for the distribution based on preset α values. - considered part of rejection region - will change based on type of hypothesis testing - found in tables - doesn't need to be calculated

critical values

1.65, 1.96, 2.33, 2.58 is referred to as

critical values when giving α

2 types of conclusions

fail to reject or reject the null hypothesis

set of values for the test statistic which prohibit the null hypothesis from being rejected

fail to reject region

p > α

fail to reject the null hypothesis

this distribution is approximately normal when?

given certain values for p large samples for p̂

1.65, 1.96, 2.33, 2.58 will be used as multipliers for problems involving?

mean and the proportion of a population

statistical technique we can be confident out interval contains

mean µ

we refer to (90% ,98% ,99%) these z scores values as

multipliers / critical values

"sufficiently large"

n = 51

central limit theorem (CLT)

normal distribution n ≥ 30 µx̄ = µ σx̄ = σ/ (√n)

for valid confidence intervals for p

np̂ and nq̂ ≥ 15

hypothesis being tested. it's initially assumed to be accurate, usually researches wish to challenge or refute

null hypothesis - Ho: µ = 0 - Ho: µ ≤ 500 - Ho: p = .154

p value represents

observed significance level

random quantities for z-distribution

only x̄

Sampling distribution for p̂: -the mean for p̂ is

p

describes a population and are based on sample statistics

paramater

p < α

reject the null hypothesis

set of values for the test statistic which allows for rejection of the null hypothesis

rejection region

upper tail

right tailed

the t-table is a

right tailed table

is computed using sample data. is sometimes referred to as a point estimate

statistics

sample sizes smaller than 51 use?

t distribution

two-tailed test: - t-test

take α/2 and df and use both POSITIVE AND NEGATIVE

two-tailed test: - proportion / z-test

take α/2 to find the value at BOTTOM of table and use both POSITIVE AND NEGATIVE

if p is LESS THAN α it is likely that?

the alternative hypothesis is true

is a test procedure for determine the legitimacy of a claim to decide between 2 separate and competing claims regarding a population characteristic. - test is conducted using sample data

test of hypothesis

Fail to reject the null: - right tailed

test statistic < critical value

Fail to reject the null: - left tailed

test statistic > critical value

Fail to reject the null: - two tailed

test statistic > negative critical value test statistic < positive critical value

Reject the null: - left tailed

test statistic ≤ critical value

Reject the null: - two tailed

test statistic ≤ negative critical value test statistic ≥ positive critical value

Reject the null: - right tailed

test statistic ≥ critical value

if p is GREATER THAN α it is likely that?

the null hypothesis is true

if the null hypothesis is true the p- value represents

the probability of observing a test-statistic computed form the sample data

what comparison determines if you fail to reject or reject the null hypothesis?

the test statistic and the critical values

α is referred to as

the total area under the curve in 1 or both tails

as n --->∞ the t distribution becomes the

the z distribution

two types of error

type I error (reject null) type II error (do not reject null)

occurs when the researcher (not at fault) reject the null hypothesis when the null is accurate. α (level of significance) represents the probability of committing this error

type I error (rejects null)

occurs when researcher (not at fault) fails to reject the null hypothesis when the null should be rejected. the probability of committing this error is referred to as β and usually not known. - as α increases β decreases

type II error (fails to reject null)

where the area (rejection region) is located will change based on?

type of test - left tailed -right tailed - two tailed

Sampling distribution for p̂: -p is an

unbiased estimator for p

the true mean µ exists but it is ______

unknown

which statistic will be used to determine an interval estimate for the population mean µ?

use the sample mean x̄ to find an interval estimate for the population mean µ x̄ ----> µ

which statistic will be used to determine an interval estimate for the population proportion p?

use the sample proportion p̂ to find an interval estimate for the population proportion p p̂ -----> p

right-tailed test: - proportion / z-test

use α only to find the value at BOTTOM of table and keep POSITIVE

left-tailed test: - proportion / z-test

use α only to find the value at the BOTTOM of table and make NEGATIVE

higher the level of confidence

wider the interval

random quantities for t-distribution

x̄ and sigma (s or σ)

population proportion (p) critical values are found the same as?

z - critical values - bottom of table

sample sizes larger than 51 use?

z distribution

95% confident

z-score: ± 1.96

98% confident

z-score: ± 2.33

99% confident

z-score: ± 2.58

90% confident

z-score:± 1.65

Sampling distribution for p̂: -standard deviation for p̂ is

√p(1-p) / n