Basics of One-sample mean test
You might have learned
about hypothesis testing. In this article, you will learn how to perform the one-sample
mean test. In this type of hypothesis testing, the population mean (µ) is the
parameter of interest. We collect the sampled data from the population and make inferences and conclusions based on this sample. Before going to perform
one-sample mean testing you, need to understand the assumptions that should be
satisfied to carry out hypothesis testing.
Assumptions for one-sample mean testing
- The data should be continuous.
- The observations in the data should be independent.
- The data should come from the normal distribution.
If the data follow these
conditions, then one can go to test the claim about population mean (µ). You
should check if the population standard deviation (σ) is known before going to
test the claim. If you know, the population standard deviation in advance then
uses a one-sample Z test. Moreover, if you do not know the population standard
deviation then use a one-sample t-test.
Let us see what the
steps are involved in the one-sample mean test one by one.
Step 1: Fix the null and alternative hypotheses
The null and
alternative hypotheses for a Z test are as follows.
Step 2: Chose the correct formula to calculate the test statistics
Use the following
formula to find the test statistic for the Z test.
Use the following
formula to find the test statistic for the t-test.
In the above formulas,
M is the sample mean; µ0 is the value of mean for which we perform
the test; σ is the population standard deviation, s is the sample standard
deviation and n is the sample size.
Step 3: Fix the
significance level (α)
Most of the time the
value of α is given in the question. If you do not know the value of α then
just take α=0.05.
Step 4: Find the critical value/s or p-value
The critical values or
p-value are essential in making the decision, whether to reject or fail to
reject the null hypothesis. You can use either the statistical tables or technology like excel or TI84 calculator or R-Studio, etc. to find critical
values as well as p-value.
Step 5: Make the decisions based on critical value/ p-value
There are basically two
ways to make decisions for the hypothesis testing based on critical values or p-value.
Based on critical values
Using critical values,
you can decide the rejection region. If the value of test statistic found in
step 2 lies in the rejection region, then reject the null hypothesis at α level
of significance, otherwise fail to reject the null hypothesis.
Based on p-value
If the p-value ≤ α then
reject the null hypothesis otherwise fail to reject the null hypothesis.
Step 6: Conclusion based on the decision
If you reject the null hypothesis then
write the conclusion is as follows.
There is sufficient evidence to support the claim
the population mean is different from µ0
(OR the population mean is less than µ0 OR the
population mean is greater than µ0).
If you fail to reject the null hypothesis
then write the conclusion is as follows.
There is insufficient evidence to support the claim
the population mean is different from µ0
(OR the population mean is less than µ0 OR the
population mean is greater than µ0).
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