Download Hypothesis Test on mu

Statistics 211/Labpack/Hypothesis Tests for µ
Hypothesis Test for µ:
Goals of the lab:
1. To have students understand the concept of a hypothesis test.
2. To have students be able to carry out a hypothesis test of a population mean.
Background:
It is often of interest to determine whether a particular value for a parameter is plausible based upon our
sample. This is different from what we did by making a confidence interval. There we determined a range
of plausible values. Here we are determining whether or not a particular value is plausible. For a twosided hypothesis test, with alpha the same, hypothesis testing and confidence interval estimation yield
equivalent results.
The idea of a hypothesis test is this. We start by assuming that the hypothesized parameter value is correct.
We then determine how far the sample mean (computed using the observed data) is from the hypothesized
value. If the sample mean and the hypothesized parameter value are "far enough" apart, then we reject the
hypothesized value. If the sample mean and the hypothesized mean are not "far enough" apart then we do
not reject H0. This idea of "far enough" is determined (in part) by the significance level, alpha.
Relevant Formula:
If
1) s is unknown AND n>2 and the data is normal
OR
2) n ≥ 30
then we can use the following to test a particular value for the population mean.
Two-sided alternative hypothesis:
1. H0: m = m0
2. HA: m ≠ m0
3. Significance level is a.
4. The test statistic is
t=
X - m0
.
Ê sx ˆ
Ë
n¯
5. Determine the p-value of the test statistic. The p-value is the probability that you get a result that is as
extreme or more extreme than the result you observed, i.e., 2*P(T > | test-stat. value |). The p-value will
come from a t-dist. with n-1 d.f. If n is large we can approximate the p-value by using the standard normal
distribution.
6. Decision Rule: Reject H0 if 2*P(T > | test-stat. value |) ≤ a. .
Do not reject H0 if 2*P(T > | test-stat. value |) > a. .
7. State conclusion.
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Statistics 211/Labpack/Hypothesis Tests for µ
One-sided alternative hypothesis:
1. H0: m = m0
2. HA: m > m0
3. Significance level is a.
4. The test statistic is
t=
X - m0
.
Ê sx ˆ
Ë
n¯
5. Determine the p-value of the test statistic. The p-value is the probability that you get a result that is as
extreme or more extreme than the result you observed, i.e., P(T > test-stat. value ). The p-value will come
from a t-dist. with n-1 d.f. If n is large we can approximate the p-value by using the standard normal
distribution.
6. Decision Rule: Reject H0 if P(T > test-stat. value) ≤ a..
Do not reject H0 if P(T > test-stat value) > a..
7. State conclusion
One-sided alternative hypothesis:
1. H0: m = m0
2. HA: m < m0
3. Significance level is a.
4. The test statistic is
t=
X - m0
.
Ê sx ˆ
Ë
n¯
5. Determine the p-value of the test statistic. The p-value is the probability that you get a result that is as
extreme or more extreme than the result you observed, i.e., P(T < test-stat. value ). The p-value will come
from a t-dist. with n-1 d.f. If n is large we can approximate the p-value by using the standard normal
distribution.
6. Decision Rule: Reject H0 if P(T < test-stat. value) ≤ a..
Do not reject H0 if P(T < test-stat. value) > a.
7. State conclusion
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Statistics 211/Labpack/Hypothesis Tests for µ
Example:
Consider a one-sided hypothesis test of m = 50 with a = 0.05. Suppose that n = 45,
X = 53.42, sx = 8.45.
Since n>30, we can use the hypothesis test, described above.
The hypothesis test goes as follows:
1. H0: m = 50
2. H1: m > 50
3. Significance level is a.=0.05
4. The test statistic is
t=
X - m0 53.42 - 50
.=
= 2.715
Ê sx ˆ
Ê 8.45 ˆ
Ë
n¯
Ë 45 ¯
5. The p-value of the test stat. is P(T > 2.715) ª 0.0034
6. Decision Rule: Reject H0 if P(T > | t-stat. value |) ≤ a. .
Since the P(T > | t-stat. value | ) ≤ a.
i.e., since 0.0034 ≤ 0.05,
we reject the null hypothesis that the population mean is 50 at the 0.05 significance level.
7. We conclude that, at the 0.05 significance level, the population mean is significantly greater than 50.
There is sufficient evidence in the sample to suggest that the population mean is greater than 50.
.
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