Download Chapter 7 Definitions, Basic Concepts

STT 315 Chapter 7 Notes
Chapter 7: Inferences Based on a Single Sample: Test of Hypotheses
Section 1: The Elements of a Test of Hypothesis
Section 2: Formulating Hypotheses and Setting up the Rejection Region
Definition: A statistical hypothesis is a statement about the numerical value of a population parameter.
Definition: The null hypothesis, denoted H0 , represents the hypothesis that will be ”retained” unless
the data provide convincing evidence that it is false. This usually represents the ”status quo.”
You may think of the null hypothesis as the ”favored” hypothesis; we reject it in favor of the alternative hypothesis Ha if and only if the evidence provided by the sample data is strongly against H0 and in favor of Ha .
Definition: The alternative (research) hypothesis, denoted Ha , represents the hypothesis that will
be accepted only if the data provide convincing evidence of its truth. This usually represents the values of a
population parameter for which the researcher wants to gather evidence to support. It is sometimes called
the research hypothesis.
Definition: The test statistic is a sample statistic, computed using the sample data, that the researcher
uses to decide between the null and alternative hypotheses.
Definition: A Type I error occurs if the researcher rejects the null hypothesis in favor of the alternative hypothesis when, in fact, H0 is true. The probability of committing a Type I error is denoted by α.
Type I error is usually thought of as the more serious error.
Definition: A Type II error occurs if the researcher retains the null hypothesis when, in fact, H0 is
false (Ha is true). The probability of committing a Type II error is usually denoted by β. Type II error is
usually thought of as the less serious error.
Steps for Selecting the Null and Alternative Hypotheses
1. Select the alternative hypothesis as that which the sampling experiment is intended to establish. The
alternative hypothesis will assume one of three forms:
a. One-tailed, upper-tailed (Ha : µ > µ0 )
b. One-tailed, lower-tailed (Ha : µ < µ0 )
c. Two-tailed (Ha : µ 6= µ0 )
2. Select the null hypothesis as the status quo, that which will be presumed true unless the sampling experiment conclusively establishes the alternative hypothesis. The null hypothesis will be specified as that
parameter value closest to the alternative in one-tailed tests and as the complementary (or only unspecified)
value in two-tailed tests.
Definition: A one-tailed test of hypothesis is one in which the alternative hypothesis Ha is directional and
includes the symbol ” < ” or ” > ”.
A two-tailed test of hypothesis is one in which the alternative hypothesis Ha does not specify departure
from H0 in a particular direction and is written with the symbol ” 6= ”.
Definition: The rejection region of a statistical test is the set of possible values of the test statistic
for which the researcher will reject H0 in favor of Ha .
1
Section 3: Observed Significance Levels: p-Values
Definition: The observed significance level, or p-value, for a specific statistical test is the probability
(assuming H0 is true) of observing a value of the test statistic that is at least as contradictory to the null
hypothesis, and supportive of the alternative hypothesis, as the actual one computed from the sample data.
How to Decide Whether to Reject H0 based on the p-value ?
1. Choose the maximum value of α you are willing to tolerate (or simply use the one given in the problem)
2. If p-value < α, reject the null hypothesis.
If p-value ≥ α, do not reject the null hypothesis.
Section 4: Test of Hypothesis about a Population Mean: Normal (z) Statistic
Conditions Required for a Valid Hypothesis Test for µ using a Z-statistic:
1. A random sample is selected from the target population.
2. X̄ has an approximately normal distribution. This can be guaranteed if:
a. The sample size n is large (that is, n ≥ 30).
b. The distribution of the original population is approximately normal.
3. The population standard deviation, σ, is known.
The test statistic is given by:
Zobs =
X̄ − µ0
√σ
n
where
X̄ is the sample mean
µ0 is the value of the population mean under the null hypothesis
σ is the standard deviation of the population
n is the sample size
The test statistic has a standard normal distribution; that is, N(0, 1).
Section 5: Test of Hypothesis about a Population Mean: Student’s t-Statistic
Conditions Required for a Valid Hypothesis Test for µ using a T-statistic:
1. A random sample is selected from the target population.
2. X̄ has an approximately normal distribution. This can be guaranteed if:
a. The sample size n is large (that is, n ≥ 30).
b. The distribution of the original population is approximately normal.
3. The population standard deviation, σ, is unknown.
We use the sample standard deviation s instead.
The test statistic is given by:
Tobs =
X̄ − µ0
√s
n
where
X̄ is the sample mean
µ0 is the value of the population mean under the null hypothesis
s is the standard deviation of the sample
n is the sample size
The test statistic has a T-distribution with n − 1 degrees of freedom.
2
Section 6: Large-Sample Test of Hypothesis about a Population Proportion
Conditions Required for a Valid Large-Sample Hypothesis Test for p:
1. A random sample is selected from a binomial population.
2. The sample size n is large. This condition will be satisfied if both np0 ≥ 15 and n(1 − p0 ) ≥ 15.
Note that np0 and n(1 − p0 ) are simply the expected number of successes and expected number of failures,
respectively, in the sample.
The test statistic is given by:
Zobs =
p̂ − p0
p̂ − p0
where
=q
σp̂
p0 (1−p0 )
n
p̂ is the sample proportion
p0 is the value of the population proportion under the null hypothesis
n is the sample size
The test statistic has a standard normal distribution; that is, N(0, 1).
3