Chapter 11 Introduction To Hypothesis Testing Flashcards
Hypothesis testing
Determining whether there is enough statistical evidence to conclude a hypothesis about a parameter is supported by the data
Tests two hypotheses:
Null hypothesis H0: default “conclusion”
Alternative or research hypothesis H1: thing you’re trying to determine if there is evidence to support
If sufficient evidence to support H1: reject the null hypothesis in factor of the alternative
If lack sufficient evidence: not rejecting the null hypothesis
Type I error
Rejecting a true null hypothesis
Probability denoted by alpha
Alpha is inverse of beta (1-beta = alpha)
Probability of type I error: significance level
Type II error
Not rejecting a false null hypothesis
Probability denoted by beta
Inverse of alpha (1-alpha = beta)
Null hypothesis form
Always states that the parameter equals the value specified in the alternative hypothesis
So always H0=
Test statistic
Statistic calculated from a random sample of the population on which to test a hypotheses
Statistic should be best for estimate of the parameter
If test statistic’s value is inconsistent with the null hypothesis we reject the null hypothesis and infer that the alternative hypothesis is true
Rejection region
A range of values such that if the test statistic falls into that range we decide to reject the null hypothesis in favor of the alternative hypothesis
For standardized test statistic the rejection region is all values of z greater than za, where a is the significance level (probability estimation method will be wrong)
Direction of inequality in the rejection region (z< or > za) matches the direction of the inequality in the alternative hypothesis
If positive za rejection region is in the right tail, negative za rejection region in the left tail
Significance level
How frequently the estimating procedure will produce an answer that will be wrong
For determining the rejection region for hypothesis testing the significance level is the a in Z>Za
Standardized test statistic
Z= (mean of x minus population mean)/ (standard deviation / sqroot of n)
Statistically significant
When a null hypothesis is rejected the test is said to be statistically significant at the significance level at which the test was conducted
P- value
The p-value of a test is the probability of observing a test statistic at least as extreme as the one computed given that the null hypothesis is true
Calculate p(xbar > specific value of xbar)
- convert xbar and value to z format to get p(z>result)
- Use table to find probability z < result and subtract to get complement
Interpreting the p value
Says that the probability of observing a statistic of x from a population whose parameter is the null hypothesis value is the p value
NOT the probability the null hypothesis is true. (Cannot make a probability statement about a parameter)
Smaller p value = more support for alternative hypothesis
Describing the p value
P-value < 0.01 = test is highly significant (overwhelming evidence the alternative hypothesis is true)
- 01 < P-value < 0.05 = test is significant (there is strong evidence the alternative hypothesis is true)
- 05 < P-value < 0.1 = test is not statistically significant (there is weak evidence that the alternative hypothesis is true)
P-value > 0.1 = little to no evidence to infer alternative hypothesis is true
Conclusions of a test of hypothesis
If we reject the null hypothesis, we conclude that there is enough statistical evidence to infer that the alternative hypothesis is true.
If we do not reject the null hypothesis, we conclude that there is not enough statistical evidence to infer that the alternative hypothesis is true. (If the value of the test statistic does not fall into the rejection region)
Never actually proving alternative hypothesis is true via statistical inference
One-tail tests
When the rejection region is located in only one tail of the sampling distribution
P value computed by finding the area in one tail of the distribution
Alternative hypothesis specifies greater than : right tail
Alternative hypothesis specifies less than : left tail
Two-tail tests
Conducted when the alternative hypothesis specifies the mean is not equal to the value stated in the null hypothesis
Requires looking at both the left and right tails of the sampling distribution
This two-tail rejection region requires that a be divided by two (because total area in rejection region must = a)
Because z< -za or z> za