Session 15 Notes, Section 8-3
Hypothesis Testing: Z Tests for Means
At the end of this chapter you will be able to:
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The z test is a statistical test for the mean of a population. It can be used
when n. or = 30, or when the population is normally distributed and sigma
(population standard deviation) is known.The formula for the z test is:
The central limit theorem states that when the population standard deviation
of σ is unknown, the
sample standard deviation s can be used in
the formula as long as the sample size is 30 or more.
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A researcher reports that the average salary of assistant professors is more
than $37000. A sample of 24 assistant professors has a mean salary of $38220. At
α = 0.05, test the claim that the assistant
professors earn more than $37000 a year. The standard deviation of the
population is $5140. Do we reject H0?
- Review the steps for hypotheses testing.
Follow them step by step.
- Step 1: State the hypotheses and identify the claim.
- H0: µ ≤ $37000 and H1: µ >
$37000 (claim)
- Step 2: Find the critical value.
- Since = 0.05 and the test is right-tailed test, the critical value is z
= +1.65.
- Step 3: Compute the test value using
z = (X -
μ) /(
σ/√n)
= +1.16.
- Step 4: Make the decision.
- Since the test value, +1.16, is less than the critical value, +1.65, and
is not in the critical region, the decision is not to reject the null
hypothesis.
- Step 5: Summarize the results.
- There is not enough evidence to support the claim that assistant
professors earn more on average than $37000 a year.
- We do not reject H0.
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A medical foundation reports that the average cost of rehabilitation for
stroke victims is $24139. To see if the average cost of rehabilitation is
different at a particular hospital, a researcher selected a random sample of
37 stroke victims at the hospital and found that the average cost of their
rehabilitation is $25153. The standard deviation of the population is $3257.
At α= 0.10, can it be concluded that the
average cost of stroke rehabilitation at a particular hospital is different
from $24139?
- Review the steps for hypotheses testing.
Follow them step by step.
- Step 1: State the hypotheses and identify the claim.
- H0: µ = $24139 and H1: µ not =
$24139
- Step 2: Find the critical value.
- Since = 0.10 and the test is a two-tailed test, the critical values are
+1.65 and -1.65.
- Step 3: Compute the test value using
z = (X -
μ) /(
σ/√n).
-
when you put in the values you get:
z = (25153 -
24139) /( 3257/√37)
or 1.89
- Step 4: Make the decision.
- Since the test value, 1.89, is greater than the critical value, +1.65 ,
and it falls in the noncritical region, the decision is to reject the null
hypothesis.
- Step 5: Summarize the results.
- There is enough evidence to support the claim that the average cost of
rehabilitation at the particular hospital is different from $24139.
- We do reject H0.
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-
- The researcher should distinguish between
statistical significance and practical
significance.
- When the null hypothesis is rejected at a specific significance level, it
can be concluded that the difference is probably not due to chance and thus is
statistically significant. However, the results may or may not have any
practical significance.
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- Step 1 State the hypothesis, and identify the claim.
- Step 2 Find the critical value from the appropriate table.
- Step 3 Compute the test value using
z = (X
-
μ) /(
σ/√n).
- Step 4 Make the decision to reject or not reject the null hypothesis.
- Step 5 Summarize the results.
Return to objectives
- The z test is used when the population standard deviation is known and the
variable is normally distributed or when σ
is not known and the sample size is greater than or equal to 30.
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