Ch. 7 and 8 Test

General Significance Testing Procedure

1. State null and alternative hypotheses, and the significance level (alpha) that is going to be used 2. Carry out the experiment, collect the data, verify the assumptions, and if appropriate compute the value of the test statistic 3. Calculate the p-value (or rejection region) - P(Z < Zobs) less than - P(Z > Zobs) greater than - 2P (Z > |Zobs|) not equal to 4. Make a decision on the significance of the test (reject or fail to reject H0) - If p-value is less than alpha, reject H0; if greater than alpha fail to reject 5. Make a conclusion statement in the words of the original problem - P-value < x -> reject H0 -> sufficient - P-value > x -> fail to reject H0 -> insufficient

Most common Values of Confidence

90%, 95%, 98%, and 99%

Significance level

Compute p-value of the test and compare directly to the significance level; Create a rejection region and compare the test statistic to this rejection region to make a decision

Population proportion

Greek letter "pi"

Calculator steps for point estimate (p-hat)

STAT, TESTS, A:1-PropZInt, X is number of successes & n is sample size, enter required components and hit Calculate

Calculator steps for significance test

STAT, TESTS, for confidence intervals choose A:1-PropZInt and for tests choose 5:1-PropZTest, enter required info and hit calculate

Assumptions

The data being used to make inferences must be a simple random sample from the population (SRS) -> put check mark if correct; The population distribution must be known to be normal, or the sample size must be "large enough" for the central limit theorem to apply -> large sample for CLT; check if n*pi is greater than or equal to 10 and n*(1-pi) is greater than or equal to 10

Confidence Interval - Interpretation

We have (x)% confidence that (parameter) is between (lower end of interval and upper end); M.O.E. -> small; C.L. -> high

Calculation of p-value requires we answer two questions:

What is the distribution of the test statistic? (Either standard normal (Z) of student's t); What type of test is being conducted? (Depends on the sign in the alternative hypothesis; greater than sign -> upper one-sided; less than sign -> lower one-sided; not equal to -> two-sided test)

Probability

Z-score can be used to convert to a standard normal distribution; Z = p-hat - mean/s.d.; Then use a normal table to make probability statements

Confidence Interval Steps

begin by selecting a sample from population, then collect necessary information from those in sample, data collected is used to compute a statistic and becomes starting point for confidence interval and hence statistical inference; if sample is representative of population, then point estimate will be a good estimate of population parameter and hence would be very close to actual value of parameter

Null hypothesis

denoted by H0; is a conjecture about population parameter that is presumed to be true; usually a statement of no effect or no change

Alternative (or research) hypothesis

denoted by Ha or H1; is a conjuncture about a population parameter that the researcher suspects or hopes it true

Sampling distribution

distribution of values taken by the statistic in a large number of simple random samples of the same size n taken from the same population

Two Types of Statistical Inferences

estimation of parameters using confidence intervals and statistical tests

Inference

first step in any inference procedure is to state the practical question that needs to be answered; involves specifying population of interest and then the specific parameter that inferences need to be made about

Statistical Inference

involves using statistics computed from data collected in a sample to make statements (inferences) about unknown population parameters

Significance level

maximum probability of a type I error that researcher is willing to risk; usually denoted by Greek letter alpha and is set to be a small probability (.01, .05, or .10); always stated at the beginning of the process (same time two hypotheses are stated)

Center

mean of p-hat is pi

Sample proportion

p-hat

P-value

probability, assuming null hypothesis is true, that test statistic takes a value as extreme or more extreme than the value observed

Type I Error

rejecting null hypothesis when it is actually true; probability of making error goes down when probability of making Type II error goes up

Description using

shape, center, spread, and unusual features

Test statistic

some quantity calculated from sample data that we have collected; used to determine strength of evidence against null hypothesis; if close to hypothesized value then likely that null hypothesis is correct

Spread

standard deviation of p-hat is square root of pi(1-pi)/n; p-hat ~ N((mean (pi), s.d.))

Confidence Intervals

statistical procedures that allow for the estimation of unknown population parameters

Margin of Error

subtracted and added to a quantity to create an interval of values in which we hope the parameter will be contained

Random variable X values:

success (1; pi) or failure (0; 1-pi)

Reject H0

sufficient evidence to conclude that alternative hypothesis is true; test is said to be significant

Fail to reject H0

we do not have sufficient evidence to conclude that the alternative hypothesis is true; test is said to be insignificant

Test of Significance

with statistical tests, we conjecture that the unknown population parameter equals some value (referred to as statistical hypothesis) and then we use the data in the sample to test whether this value is reasonable or not