Steps in Hypothesis Testing
1.State H0 and H1.
2.Set alpha ().
3.Gather data for the test.
4.Calculate the test statistic.
5.Use a Decision Rule or P-value to reach a
conclusion.
Z Test of Hypothesis for the Mean ( known)
(Sections 9.2 and 9.3 of the text)
(Use Z-TEST FOR THE MEAN, SIGMA KNOWN in
the EXCEL addin PHSTAT)
1.H0:  = 0
H1:   0
2. Set alpha ().
3. Gather data for the test.
4. Test Statistic Z = ( X - 0)/ / n
5. Look up Z for /2 in Z table. If the absolute value of the
test statistic Z is greater than the Z from the table or [Pvalue < ], reject H0; that is, there is evidence that the
population has different mean than hypothesized (0). If
you don’t reject H0, assume the mean is as hypothesized. *
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 is the true mean in the population.
0 is the hypothesized mean in the population (Note: your text just uses the
symbol  in the step 4 equation).
X is the mean in a sample from the population.
 is the true standard deviation in the population.
n is the number in the sample taken from the population.
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* To convert to one tail, look up Z for . Then if upper tail, check if the test
statistic Z is greater than the Z from the table. If lower tail, check if
the test statistic Z is less than the minus Z from the table.
t Test of Hypothesis for the Mean ( unknown)
(Section 9.4 of the text)
(Use T-TEST FOR THE MEAN, SIGMA UNKNOWN
in the EXCEL addin PHSTAT)
1.H0:  = 0
H1:   0
2. Set alpha ().
3. Gather data for the test.
4. Test Statistic t = ( X - 0)/ s/ n
5. Look up t (column = /2, row = n-1) in t table. If the
absolute value of the test statistic t is greater than the t from
the table, reject H0; that is, there is evidence that the
population has different mean than hypothesized (0). If
you don’t reject H0, assume the mean is as hypothesized. *
************************************
 is the true mean in the population.
0 is the hypothesized mean in the population.
X is the mean in a sample from the population. (Note: your text
just uses the symbol  in the step 4 equation).
s is the estimate of the standard deviation in a sample from the
population.
n is the number in the sample taken from the population.
************************************
* To convert to one tail, look up t for  and row n-1. Then if upper tail,
check if the test statistic t is greater than the t from the table. If lower
tail, check if the test statistic t is less than the minus t from the table.
Z Test of Hypothesis for the Proportion
(Section 9.5 of the text)
(Use Z-TEST FOR THE PROPORTION in the EXCEL
addin PHSTAT)
1.H0:  =  0
H1:    0
2. Set alpha ().
3. Gather data for the test.
4. Test Statistic Z = (p -  )/ Square Root (  0 (1-  0)/n).
5. Look up Z for (1- /2) in Z table. If the absolute value of
the test statistic Z is greater than the Z from the table or [Pvalue < ], reject H0; that is, there is evidence that the
population has a different proportion than hypothesized
(  0). If you don’t reject H0, assume the population
proportion is as hypothesized. *
0
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 is the true proportion in the population.
 0 is the hypothesized proportion in the population. (Note: your
text just uses the symbol  in the step 4 equation).
p is the proportion as estimated from a sample of the population.
n is the number in the sample taken from the population.
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* To convert to one tail, look up Z for . Then if upper tail, check if the test
statistic Z is greater than the Z from the table. If lower tail, check if
the test statistic Z is less than the minus Z from the table.
To Determine Which T-Test to Use,
First use this test:
F Test for Differences in Two Variances
(Be sure the first variable has the largest variance)1 (section xx)
(Use F TEST TWO SAMPLE FOR VARIANCES in TOOLS
in EXCEL)
1.H0: 12 = 22
H1: 12  22
2. Set alpha ().
3. Gather data for the test.
4. Test statistic F = s12 /s22.
5. If [F > F (page = /2, column=n1-1, row = n2-1)] or [Pvalue < ], reject H0; that is, assume variances are not
equal. If you don’t reject H0, assume the variances are
equal.
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1 and 2 are the standard deviations in population 1 and 2.
s1 and s2 are the standard deviations in sample 1 and 2.
n1 and n2 are the number in sample 1 and 2.
1
Note: This is a little different than the text. But by putting the largest variance first it is
easier to look up the F statistic and you only have to look up one F statistic rather than
two. Also then the procedure is like that used in the EXCEL addin.
Pooled-Variance T Test for Differences in
Two Means (It assumes the variances are equal)
(Often called the pooled variance t-test) (Text Section xx)
(Use T-TEST TWO SAMPLE ASSUMING EQUAL
VARIANCE in TOOLS in EXCEL )
1. H0: 1 - 2 = d
H1: 1 - 2  d
2. Set alpha ().
3. Gather data for the test.
4. t = (( X 1 - X 2)- d )/square root (sp2/n1+sp2/n2 )
where
sp2=[(n1-1)s12 + (n2-1)s22]/[ n1+ n2 -2]
5. Look up t (column = /2, row = n1 + n2 - 2) in t table. If
[- table t < t < + table t] or [P-value < ], reject H0. That is,
there is evidence that the groups have different means. If
you don’t reject H0, assume the means are equal.
************************************
1 and 2 are the true means in population 1 and 2.
d is the hypothesized difference between the means in population
1 and 2. (This is usually 0.)
X 1 and X 2 are the means in sample 1 and 2.
sp2 is the pooled variance from combining groups 1 and 2
s1 and s2 are the standard deviations in sample 1 and 2.
n1 and n2 are the number in sample 1 and 2.
T Test for Differences in Two Means
(It assumes the variances are NOT equal)
(Not fully explained in section xx in the text)
(Use T-TEST TWO SAMPLE ASSUMING UNEQUAL
VARIANCE in TOOLS in EXCEL)
1.H0: 1 - 2 = d
H1: 1 - 2  d
2. Set alpha ().
3. Gather data for the test.
4. Test Statistic t = (( X 1 - X 2)- d )/ square root (s12/n1+s22/n2).
5.Look up t (column = /2, row = df) in t table. If [- table t
< t < + table t] or [P-value < ], reject H0. That is, there
is evidence that the groups have different means. If you
don’t reject H0, assume the means are equal. In the table
lookup, the row number must be revised (from n1 + n2 2) due to unequal variances; calculate it as follows:
df=[s12/n1+s22/n2 ]2/[((s12/n1)2/(n1-1)) + ((s22/n2 )2/(n2-1))]
************************************
1 and 2 are the true means in population 1 and 2.
d is the hypothesized difference between the means in population
1 and 2. (This is usually 0.)
X 1 and X 2 the means in sample 1 and 2.
s1 and s2 are the standard deviations in sample 1 and 2.
n1 and n2 are the number in sample 1 and 2.
F Test for Differences in C Means
(Often called Analysis of Variance F Test or ANOVA)
(Use ANOVA: SINGLE FACTOR in TOOLS in EXCEL)
(Section 10.5)
1.H0: 1 = 2 =... = c
H1: at least one mean is different.
2. Set alpha ().
3. Gather data for the test.
4. Test statistic F = (MS Between)/(MS Within)
> F table value (page = , column=c-1, row = nc)] or [P-value < ], reject H0 . That is, there is evidence
that the groups have different means. If you don’t reject
H0, assume the means are equal.
If you find that there is a difference between groups using
ANOVA, find the right Q in (table = , column =c and row
n-c) and use the TUKEY-KRAMER in PHSTAT to find
which groups are different.
5. If [F
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i
is the mean in population i.
c
is the number of different groups.
n
is the total number of data points in all samples combined.
MS Between is the Mean Squared Error Between Groups
MS Within is the Mean Squared Error Within Groups
Z Test for the Difference in Two Proportions
(Text Section xx)
(Use Z-TEST FOR THE DIFFERENCES IN TWO
PROPORTIONS in PHSTAT IN EXCEL )
1.H0: p1 - p2 = pd
H1: p1 - p2  pd
2. Set alpha ().
3. Gather data for the test.
4. Z = ((ps1 - ps2) - pd)/ square root ( p (1- p ) (1/n1+1/n2 ))
where
p = (n1 ps1 + n2 ps2)/ (n1 + n2)
5. Look up Z for (1- /2) in Z table. If [- table Z < Z < +
table Z] or [P-value < ], reject H0. That is, there is
evidence that the groups have different proportions. If you
don’t reject H0, assume the proportions are equal.
************************************
p1 and p2 are the true proportions in population 1 and 2.
pd is the hypothesized difference between the proportions in
population 1 and 2. (This is usually 0.)
ps1 and ps2 are the proportions in sample 1 and 2.
p is the average proportion in sample 1 and 2 combined.
n1 and n2 are the number in sample 1 and 2.
Chi Squared (2) Test for Differences in c
Proportions
(Often just called chi squared test)(Text Section 11.2)
(Use CHI SQUARED TEST under C SAMPLE TESTS in the
PHSTAT addin for EXCEL. Do that first, then enter the number of
rows and columns of data and finally add the values in each cell.)
1.H0:  1 =  2 =... =  c
H1: at least one proportion is different.
2. Set alpha ().
3. Gather data for the test.
4. Test statistic is 2 =  (fo-fe)2/ fe summed over all cells.
5. If [2 > 2(column = , row = (r-1)(c-1))] or [P-value <
], reject H0. That is, there is evidence that the groups have
different proportions; if so, you should be able to use the
Marasculio procedure. If you don’t reject H0, assume the
proportions are equal.
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 i is the proportion in population i.
fo is the observed frequency in each cell
fe is the expected frequency in each cell.
c is the number of columns (groups) in the cross tabulation table.
r is the number of rows in the cross tabulation table.
T Test for the Slope
(Hypothesis Test on a Regression Coefficient)
(An option to request when using Regression in the
PHSTAT addin in EXCEL).
(Text Section 12.7)
1.H0: i = 0 (no linear relationship)
H1: i  0 (a linear relationship)
2. Set alpha ().
3. Gather data for the test.
4. Test statistic t = bi / sbi
5. Look up t (column = /2, row = n-2) in t table. If the
absolute value of the test statistic t is greater than the t
from the table or [P-value < ], reject H0. That is, there
is evidence that there is a linear relationship. If you
don’t reject H0, there is not enough evidence to show
that there is a linear relationship.
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i is the actual regression coefficient in the population.
bi is the regression coefficient in the sample.
sbi is the standard error of the sample regression coefficient.
n is the number in the sample.
Test for the Significance of the Multiple
Regression Model
(Hypothesis Test on the Overall Regression)
(An option with Multiple Regression in the
PHSTATS addin in EXCEL).
(Text Section 13.4)
1.H0: 1 = 2 = .... = p = 0 (No linear
relationships at all)
H1: at least one linear relationship.
2. Set alpha ().
3. Gather data for the test.
4. Test Statistic F = (MS Regression) /(MS Residual)
5. If [F > the F in table (page = , column = p, row = n - p 1)] or [P-value < ], reject H0. That is, there is evidence
that at least one linear relationship exists. If you don’t
reject H0, there is not enough evidence to show any
linear relationships exist.
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i is the actual regression coefficient in the population.
p is the number of predictors.
n is the number in the sample.
MS Regression is the Mean Squares explained by the Regression.
MS Residual is the Mean Squares of the remaining Error.
Durbin-Watson Statistic.
(Also known as the Durbin-Watson Test)
(An option with Regression in the PHSTATS addin
in EXCEL).
(Text Section 12.6)
1. H0: The Residuals are Independent
H1: The Residuals are not Independent
2. Set alpha ().
3. Gather data for the test as part of a regression analysis.
4. Test Statistic D = (ei - e i-1)2/  ei2
5. If [ D < dl (table = , column = p (use dl subcolumn),
row = n in table E.9)], reject H0. That is, there is
evidence that the residuals are not independent. If you
don’t reject H0, assume the residuals are independent.
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ei is the error for residual i.
n is the number in the sample.
p is the number of predictors.
T-Test for Significance of Association.
(Hypothesis Test on a Correlation Coefficient)
(Not available in EXCEL).
(Text Section 12.7)
1.H0:  = 0 (No linear relationship)
H1:   0 (A linear relationship exists)
2. Set alpha ().
3. Gather data for the test.
4. Test Statistic t = r / square root ((1-r2)/(n-2))
5. Look up t (column = /2, row = n-2) in t table. If the
absolute value of the test statistic t is greater than the t
from the table, reject H0. That is, there is evidence that
there is a linear relationship. If you don’t reject H0, you
do not have enough evidence to show that there is a
linear relationship.
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 is the actual correlation coefficient in the population.
r is the correlation coefficient in the sample.
n is the number in the sample.
October 25, 2005