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