Stats Unit 6

null hypothesis

denoted by H0, is usually the hypothesis that sample observations result purely from chance.

alternative hypothesis

denoted by H1 or Ha, is the hypothesis that sample observations are influenced by some non-random cause.

Standard deviation for two sample z-test

these would be the sigmas (the standard deviations) that are provided by the problem

Standard deviation for two-sample t-test

these would be the the sample standard deviation that are given to us by the problem

two-sample t-Test

used to determine if two population means are equal. The variances of the two samples may be assumed to be equal or unequal.

two-sample z-Test

used to determine if two population means are equal. The variances of the two samples may be assumed to be equal or unequal.

matched pairs test

used when the data from the two groups can be presented in pairs, for example where the same people are being measured in before-and-after comparison or when the group is given two different tests at different times

one-sample vs. two-sample tests

A Paired Sample Tests compares two sample means from the same population regarding the same variable at two different times such as during a pre-test and post-test, or it compares two sample means from different populations whose members have been matched.

Type II Error

is incorrectly retaining a false null hypothesis

standard error

that measures the accuracy with which a sample represents a population; a sample mean deviates from the actual mean of a population

Type I Error

the incorrect rejection of a true null hypothesis

Probability of Type I Error

the level of significance of the test of hypothesis, and is denoted by *alpha*. Usually a one-tailed test of hypothesis is is used when one talks about type I error.

p-value

A small p-value (typically ≤ 0.05) indicates strong evidence against the null hypothesis, so you reject the null hypothesis. A large p-value (> 0.05) indicates weak evidence against the null hypothesis, so you fail to reject the null hypothesis.

t vs z

A t-test is used for testing the mean of one population against a standard or comparing the means of two populations if you do not know the populations' standard deviation and when you have a limited sample (n < 30). If you know the populations' standard deviation, you may use a z-test.

one-sided vs. two-sided

A two-tailed test is appropriate if the estimated value may be more than or less than the reference value, for example, whether a test taker may score above or below the historical average. A one-tailed test is appropriate if the estimated value may depart from the reference value in only one direction, for example, whether a machine produces more than one-percent defective products.

zTest

A z-test is a statistical test used to determine whether two population means are different when the variances are known and the sample size is large. The test statistic is assumed to have a normal distribution, and nuisance parameters such as standard deviation should be known for an accurate z-test to be performed.

Statistically Significant

In principle, a statistically significant result (usually a difference) is a result that's not attributed to chance. More technically, it means that if the Null Hypothesis is true (which means there really is no difference), there's a low probability of getting a result that large or larger.

Power

The power of a binary hypothesis test is the probability that the test correctly rejects the null hypothesis (H0) when a specific alternative hypothesis (H1) is true.

Probability of Power

The power of a hypothesis test is nothing more than 1 minus the probability of a Type II error.

test of significance

The significance level for a given hypothesis test is a value for which a P-value less than or equal to is considered statistically significant. Typical values for are 0.1, 0.05, and 0.01. These values correspond to the probability of observing such an extreme value by chance.

alpha-value

The significance level, also denoted as alpha or α, is the probability of rejecting the null hypothesis when it is true.

two sample vs two sided

When using a two-tailed test, regardless of the direction of the relationship you hypothesize, you are testing for the possibility of the relationship in both directions. Two-sample hypothesis testing is statistical analysis designed to test if there is a difference between two means from two different populations.

t-Test

an analysis of two populations means through the use of statistical examination; a t-test with two samples is commonly used with small sample sizes, testing the difference between the samples when the variances of two normal distributions are not known.

Choosing an alpha/significance level

choose an alpha value due to the overall importance of the statistic and or information

Confidence interval as a two sided test

it basically like... A 95% confidence interval for the mean lumber length was 8.03 feet to 8.57 feet. For our two-tailed test the hypotheses were: Since 8.5 falls within the 95% confidence interval, we cannot reject the null hypothesis at level 0.05.

degrees of freedom

n-1 the number of values in the final calculation of a statistic that are free to vary.

central limit theorem

statistical theory that states that given a sufficiently large sample size from a population with a finite level of variance, the mean of all samples from the same population will be approximately equal to the mean of the population. 15 --> 15-40 ---> 40+