Business Statistics: Chapter 10; One-Sample Tests of Hypothesis
What is Hypothesis Testing?
Hypothesis: statement about a population parameter that must be verified.
Hypothesis Testing: the procedure based on sample evidence and probability theory used to find whether hypothesis is rational or not. Reject or fail to reject hypothesis after testing.
Six-Step Procedure for Testing a Hypothesis
Step 1: State Null Hypothesis and Alternate Hypothesis, or H₀ and H₁.
Null Hypothesis can be signaled by “no chance” or “equal to” or “There is no significant difference between. . . “ or “the mean x is not significantly different from. . . “ etc.
Null Hypothesis: statement about value of a population parameter used for testing with sample data.
Alternate Hypothesis: an inference about population parameter based on sample data when null hypothesis is rejected. Equal sign is never used in alternate hypothesis.
Step 2: Select Level of Significance, the probability of rejecting null hypothesis when it is true.
.05 level is for consumer research projects, .01 for quality assurance, and .10 for political polling, usually, but can be any value between 0 and 1.
Step 3: Select Test Statistic. z and t used in this chapter, F and x² used later.
Test Statistic: a value computed from sample information used to find whether to reject or fail to reject null hypothesis.
Testing a mean, σ KNOWN; z = [(x̄ - μ) / (σ / square root of n)]
Where x̄ is sample mean, σ is population standard deviation, n is the number of observations in a sample, and μ is population mean.
When using z-value with three decimals, use t-table and assume infinite degrees of freedom
Step 4: Formulate Decision Rule. This is a specific statement about the conditions under which the null hypothesis is rejected or failed to be rejected.
Critical value: dividing point between region where null hypothesis is rejected and region where it is failed to be rejected.
To find critical value, it depends on test statistic being used.
For z-value, find critical value with t-table and infinite degrees of freedom.
Step 5: Make a decision. Compare the value of the test statistic to the value of the critical value, or decision rule.
Step 6: Interpret results.
Error Types:
Type I Error: when null hypothesis is true but rejected. Probability represented by α.
Type II Error: when null hypothesis is false but failed to be rejected. Probability represented by β.
One-Tailed and Two-Tailed Hypothesis Tests
One-Tailed Test: Rejected region is one on one tail of the curve.
H₀ less than or equal to # and H₁ more than #, or the other way around.
Two Tailed Test: rejected region is on either side of the curve.
H₀ equal to # and H₁ not equal to #. Doesn’t specify direction.
Hypothesis Testing for a Population Mean: Known Population Standard Deviation
For Either Test: Follow six steps for testing a hypothesis.
State null and alt hypothesis:
Two-tailed test: null= #, alt not = # and one-tailed test: null less than/more than or equal to #, alt more than or less than #.
To find test statistic, z = sample mean - population mean / standard deviation of population / square root of sample size, or z = [(x̄ - μ) / (σ / square root of n)]
To find decision rule, determine critical values of z. Use t-distribution table (split level of significance for two-tailed test) and assume infinite degrees of freedom.
Decision: if z is between decision rule, it is failed to reject. If not, it is rejected. Compare test statistic to critical value.
P-Value in Hypothesis Testing
P-Value: probability of the sample outcome assuming that the null hypothesis is true. Is compared to level of significance, and if it is less than α, then it is rejected.
To find P-value, use z-table and round z-value to two decimals. If null hypothesis is two-tailed, times the value on the chart by two. Or use a p-value calculator.
Interpreting Weight of Evidence Against Null Hypothesis:
.10: some evidence null hypothesis is not true
.05: strong evidence it is not true
.01: very strong evidence
.001: extremely strong evidence.
Hypothesis Testing for a Population Mean; Unknown Population Standard Deviation
Testing a Mean, α Unknown: use t-distribution for test statistic
To find test statistic, t = [(x̄ - μ) / (s / square root of n)]
Where x̄ is sample mean, s is sample standard deviation, n is the number of observations in a sample, and μ is population mean.
To find decision rule, find degrees of freedom (unlike z-statistic, don’t need to use infinite if three-decimals). Use t-table with degrees of freedom and level of significance.
Test a Hypothesis of a Population Proportion
Test of Hypothesis, One Proportion:
To find test statistic, z = (p - π) / square root of [π (1-π ) / n]
Where π is population proportion, p is sample proportion, and n is sample size
To find decision rule, use Appendix B.5, use significance level given and use row with infinite degrees of freedom.