8.1. Things to know
Two areas of Inferential Statistics: Estimation and Hypothesis Testing.
Hypothesis Testing is an area statistical inference in which one evaluates a conjecture about some characteristic of the population based upon the information contained in the random sample. Usually the conjecture concerns one of the unknown parameters of the population.
Hypothesis is a claim or statement about the population parameter.
Steps in Hypothesis Testing
Make the statistical decision: a. If decision rule is based on region of rejection: Check if test statistic falls in the region of rejection. If yes, reject the null hypothesis. b. If decision rule is based on p-value: Determine the p-value. If the p-value is less than or equal to reject the null hypothesis.
Interpret results.
Null Hypothesis: • denoted by
• the statement being tested • it represents what the experimenter
doubts to be true • must contain the condition of equality and must be
written with the symbol •
Alternative Hypothesis: • denoted by
• is the statement that must be true if the null hypothesis is false •
the operational statement or the theory that the experimenter believes
to be true and wishes to prove • is sometimes referred as the research
hypothesis
Test of Significance: • A test of significance is a problem of deciding between the null and the alternative hypothesis on the basis of the information contained in a random sample. • The goal will be to reject in favor of , because the alternative is the hypothesis that the researcher believes to be true. If we are successful in rejecting , we then declare the results to be “significant”.
Two Types of Errors:
Type I Error – the mistake of rejecting the null hypothesis when
it is true. • It is not a miscalculation or procedural misstep; it is an
actual error that can occur. • the probability of rejecting the null
hypothesis when it is true is called the significance level
• The value of is predetermined, and very common choices are
Type II Error – the mistake of failing to reject the null hypothesis when it is false. • The symbol (beta) is used to represent the probability of a type II error.
Test Statistic: • A statistic computed from the sample data that
is especially sensitive to the differences between and
• The test statistic should tend to take on certain values when is true
and different values when is true. • The decision to reject depends on
the value of the test statistic. • A decision rule based on the value of
the test statistic: Reject if the computed value of the test statistic
falls in the region of rejection.
Region of Rejection or Critical Region: • the set of all values
of the test statistic which will lead to the rejection of
Factors that determine the region of rejection: • The behavior of the
test statistic if the null hypothesis were true. • The alternative
hypothesis: the location of the region of rejection depends on the form
of
• Level of significance the smaller is, the smaller the region of
rejection.
Critical Value/s: • The value or values that separate the critical region from the values of the test statistic that would not lead to rejection of the null hypothesis. • Depends on the nature of the null hypothesis, the relevant sampling distribution, and the level of significance.
Types of Tests:
Two-tailed Test – if we are primarily concerned with deciding whether the true value of a population parameter is different from a specified value, then the test should be two-tailed. For the case of the mean,
Left-tailed Test – if we are primarily concerned with deciding whether the true value of a population parameter is less than a specified value, then the test should be left-tailed. For the case of the proportion,
Right-tailed Test – if we are primarily concerned with deciding whether the true value of a population parameter is greater than a specified value, then the test should be right-tailed. For the case of the difference of two population means,