The 5 Steps of a Statistical Hypothesis Test Calculator Facts
嚜燜he 5 Steps of a Statistical Hypothesis Test
1. State the null hypothesis, H0
2. State the alternative hypothesis, Ha
3. Determine values for the test statistic and the standardized test statistic.
4. Find the P -value. Use the P -value to decide whether or not to reject the null hypothesis.
5. Write the full sentence conclusion (result) of the hypothesis test.
Calculator Facts
(Section 7.2) If your hypothesis test is about a population mean 米, and the value of population standard deviation 考 is known or given, then use
your calculator's "z-test" to obtain the p-value
(Section 7.3) If your hypothesis test is about a population mean 米, and the value of population standard deviation 考 is unknown or not given, then
use your calculator's "t-test" to obtain the p-value
(Section 7.4) If your hypothesis test is about a population proportion (percentage) p , then use your calculator's "1-prop-z-test" to obtain the p-value
The 5 Steps of a Statistical Hypothesis Test
1. State the null hypothesis, H0
2. State the alternative hypothesis, Ha
Two-Tailed Test
Left-Tailed Test
Right-Tailed Test
Sign in the null hypothesis, H0
=
≡
≒
Sign in the alternative hypothesis, Ha
≧
<
>
Note that the null hypothesis always has an equal to (=) or a greater than or equal to (≡ ) or a less than or equal to (≒ ) sign, and the alternative hypothesis
always has a not equal to (≧ ) or a less than () sign.
3. Determine values for the test statistic and the standardized test statistic.
What is a test statistic?
De nition
The test statistic is the sample mean, sample proportion or sample variance or standard deviation (depending on which one of the parameters,
2
米, p, 考 , or 考 , you are testing).
What is a standardized test statistic?
De nition
The standardized test statistic is the number of standard deviations that your sample statistic is above or below the hypothesized mean (of the
sampling distribution).
The Empirical Rule tells us that 95% of our test statistics (and standardized test statistics) will be within 1.96 standard deviations of the mean. 5% of test
statistics are either less than -1.96 or greater than 1.96.
Formulas used for the Standardized Test Statistic
If the test is about a population mean, 米, with the value of 考 given or known, the standardized test statistic is given by the formula z
米0
=
x? ? 米0
?
?
考/﹟n
, where
is the value of the mean selected for the null hypothesis.
If the test is about a population mean, 米, with the value of 考 not given or unknown, the standardized test statistic is given by the formula t
If the test is about a population proportion, p, the standardized test statistic is given by the formula z
population proportion selected for the null hypothesis and q0
= 1 ? p0
=
^ ? p0
p
?
??
?
p0 q0
﹟
n
=
x? ? 米0
?
?
s/﹟n
, where p0 is the value of the
.
4. Find the P -value. Use the P -value to decide whether or not to reject the null hypothesis.
What is a p-value?
De nition
The P每value is the probability of getting a value of the test statistic (and standardized test statistic) that is at least as extreme as the one representing the
given sample data, assuming that the null hypothesis is true. One often "rejects the null hypothesis" when the P每value is less than the predetermined
signi cance level (汐), indicating that the observed result would be highly unlikely under the null hypothesis.
How can I nd the p-value?
Run the appropriate test (z-test, t-test or 1-prop-z-test) on your calculator and it gives you the P -value. Draw a picture of the appropriate sampling distribution.
Center the distribution at the value used in the statement of your null hypothesis. Afterwards, label the location of the sample mean or sample proportion
along the x axis. Finally, sketch the graph of the standardized sampling distribution (either the z or t distribution), and label the location of the test statistic.
Draw a vertical line at that location. Then, use the owchart below to help you sketch the correct area under the sampling distribution that represents the pvalue.
How do I use the P -value to decide whether or not to reject the null hypothesis?
If the p-val ≒
汐
then reject the null hypothesis.
If the p-val > 汐 then do not reject the null hypothesis.
What is 汐 (alpha)?
De nition
A type I (or 汐 type) error occurs if you reject the null hypothesis (conclude the sample evidence suggests it is false), when it is really true.
De nition
The level of signi cance is the maximum probability of committing a type I error. This probability is symbolized by 汐 (Greek letter alpha). That is, P(type I
error) = 汐.
As a researcher you will preselect a value for the level of signi cance, 汐 . For the problems we do in this class, you will be given a value for 汐 in the statement of the problem. If you are not
given a value of 汐 in the statement of the problem then use a 5% signi cance level (汐
= 0.05
).
5. Write the full sentence conclusion (result) of the hypothesis test. Use the wording from the ow chart given below
?
Hypothesis Test Conclusion
This means that...
Reject H0
There is convincing evidence against the null hypothesis. If H0 were true, the sample data would be very surprising.
Fail to Reject H0
There is not convincing evidence against the null hypothesis. If H0 were true, the sample data would not be considered surprising.
Parameter
test
standardized test
how to get pvalue and
book
being tested
statistic
statistic formula
standardized test stat
section
mean, ? (考 known)
x?
z=
x? ? ?
﹟
考/ n
z-test
7.2
mean, ? (考 unknown)
x?
t=
x? ? ?
﹟
s/ n
t-test
7.3
proportion (%), p
p?
p? ? p
z = p pq
1-prop-z-test
7.4
聿2 cdf
7.5
聿2 cdf
7.5
2-Samp-z-test
8.1
2-prop-z-test
8.4
n
standard deviation, 考
s2
variance, 考 2
s2
diff between means, ?1 ? ?2
x?1 ? x?2
diff between props, p1 ? p2
p?1 ? p?2
(n ? 1)s2
考2
(n ? 1)s2
聿2 =
考2
(x?1 ? x?2 ) ? (?1 ? ?2 )
q 2
z=
考1
考22
n1 + n2
聿2 =
z=
(p?1 ? p?2 ) ? (p1 ? p2 )
s
p?q?
with
p? =
x1 + x2
n1 + n2
1
n1
and
1
+
1
n2
q? = 1 ? p?
................
................
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