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One and Two-Sample Tests of Hypotheses - Coggle Diagram
One and Two-Sample Tests of Hypotheses
Statistical Hypotheses: General Concepts
"A statistical hypothesis is an assertion or conjecture concerning one or more
populations"
The Role of Probability in Hypothesis Testing: rejection means that there is a small probability of obtaining the sample
information observed when, in fact, the hypothesis is true
reject H0 in favor of H1 because of sufficient evidence in the data or
fail to reject H0 because of insufficient evidence in the data.
H0: defendant is innocent,
H1: defendant is guilty
Testing a Statistical Hypothesis
Probability of Type 1 Error: Rejection of the null hypothesis when it is true is called a type I error
Nonrejection of the null hypothesis when it is false is called a type II error.
The Probability of a Type II Error: The probability of committing a type II error, denoted by β, is impossible to compute unless we have a specific alternative hypothesis
The Role of α, β, and Sample Size: the probability of committing both types of error can be reduced by
increasing the sample size
Illustration with a continuous random Variable: The concepts discussed here for a discrete population can be applied equally well
to continuous random variables.
Important Properties of a Test of Hypothesis: The type I error and type II error are related. A decrease in the probability
of one generally results in an increase in the probability of the other.
The size of the critical region, and therefore the probability of committing
a type I error, can always be reduced by adjusting the critical value(s).
An increase in the sample size n will reduce α and β simultaneously.
he power of a test is the probability of rejecting H0 given that a specific alternative is true.
Single Sample: Tests Concerning a Single Mean
The single mean (or one-sample) t-test is used to compare the mean of a variable in a sample of data to a (hypothesized) mean in the population from which our sample data are drawn.
This is important because we seldom have access to data for an entire population. The hypothesized value in the population is specified in the Comparison value box.
One- Sample Tests Concerning Variances
To test variability, use the chi-square test of a single variance. The test may be left-, right-, or two-tailed, and its hypotheses are always expressed in terms of the variance (or standard deviation).
If a variable X is normally distributed with mean μ and variance 𝜎𝜎2, the sample variance is distributed as a Chisquare random variable with N - 1 degrees of freedom, where N is the sample size. That is
is distributed as a Chi-square random variable. The sample statistic, s
2 , is calculated as
2 is the assumed actual value of the variance under the alternative hypothesis, then the power or sample size of a
hypothesis test about the variance can be calculated using the appropriate one of the following three formulas from
