Unformatted text preview: Business Statistics (BUSA 3101)
Business
Dr. Lari H. Arjomand
[email protected] Chapter 11 Hypothesis Testing About the Differences Between Two Population Means H0 :
µ 1  µ2 = 0 Ha :
µ 1  µ2 # 0 Slide 2 Hypothesis Testing About the Differences Between Two Population Means Let µ1 equal the mean of population 1 and µ2 equal the mean of population 2.
s The difference between the two population means is µ1 µ2.
s s Let n1 be the sample size of population 1 and n2 the sample size of population 2. x2
x1 Let equal the mean of sample 1 and equal the mean of sample 2.
s Slide 3 Hypothesis Tests About µ 1 − µ 2:
σ 1 and σ 2 Known Hypotheses H 0 : µ 1 − µ 2 ≥ D0 H 0 : µ 1 − µ 2 ≤ D0
H a : µ 1 − µ 2 < D0 H a : µ 1 − µ 2 > D0
Lefttailed Righttailed H 0 : µ 1 − µ 2 = D0
H a : µ 1 − µ 2 ≠ D0
Twotailed Test Statistic (Actual z Value) z= ( x1 − x2 ) − D0
2
σ12 σ 2
+
n1 n2 Slide 4 Two cities,
Bradford and Kane
are separated only
by the Conewango
River. There is
competition
between the two
cities. The local
paper recently reported that
with a standard deviation
the mean household income
of $7,000 for a sample of
in Bradford is $38,000 with
35 households. At the 0.01
a standard deviation of
significance level can we
$6,000 for a sample of 40
conclude the mean income
households. The same
in Bradford is more?
article reported the mean
income in Kane is $35,000 Challenging Example 1
Slide 5 Step 4
State the decision rule.
The null hypothesis is
rejected if actual z is
greater than critical z
of 2.33 or if p ≤ .01
Step 1
State the null and
alternate hypotheses.
H0: µB < µK
H1: µB > µK Step 3
Find the appropriate test
statistic. Because both
samples are more than 30, we
can use z as the test statistic.
Step 2
State the level of significance.
The 0.01 significance level is
stated in the problem.
Solution Slide 6 Step 5: Compute the value of actual z and make a decision
z= $38,000 − $35,000
($6,000) 2 ($7,000) 2
+
40
35 = 1.98 Actual Z Because the actual Z = 1.98 < critical Z = 2.33, and since the pvalue = 0.0239 > α = 0.01 the decision is to
accept the null hypothesis. Thus we cannot conclude
that the mean household income in Bradford is larger
then the mean household income in Kane.
H0: µB < µK H1: µB > µK Slide 7 Hypothesis Tests About µ 1 − µ 2:
σ 1 and σ 2 Unknown and Small Sample
When σ 1 and σ 2 are unknown, we will: • use the sample standard deviations s1 and s2
as estimates of σ 1 and σ 2 , and • use t table instead of Z table.
•Assuming at least one of the sample is n < 30 Slide 8 Hypothesis Tests About µ 1 − µ 2:
σ 1 and σ 2 Unknown and Small Sample s Hypotheses H 0 : µ 1 − µ 2 ≥ D0 H 0 : µ 1 − µ 2 ≤ D0
H a : µ 1 − µ 2 < D0 H a : µ 1 − µ 2 > D0
Lefttailed
s Righttailed H 0 : µ 1 − µ 2 = D0
H a : µ 1 − µ 2 ≠ D0
Twotailed Test Statistic (Actual t)—When n < 30 and σ is unknown: t= ( x1 − x2 ) − D0
2
s12 s2
+
n1 n2 Slide 9 EXCEL APPLICATION
When in a problem raw data are given, then
you may use either Data Analysis AddIns
or SWStat+ AddIns in Excel to solve the
problem. See next slides. Hypothesis Tests About µ 1 = µ 2 When
σ 1 and σ 2 Unknown & Small Samples
Challenging Example #1
Using SWStat+ and Data Analysis in Excel
s Ariana Corporation wants to increase the
Ariana
productivity of its line workers. To do so, two
different programs have been suggested to help
increase productivity. Twenty employees,
making up a sample, have been randomly
assigned to one of the two programs and their
assigned
output for a day's work has been recorded. The
results are given on the next slide.
results Slide 11 Example #1 Continued
Program A Program B 150 150 130 120 120 135 180 160 145 110 185 175 220 150 190 120 180 130 175 220 Slide 12 Example #1 Continued
Questions
s
s s a. State the null and alternative hypotheses.
b. Use Excel, and at 95% confidence test
to determine if the means of the two
populations are equal, and
c. Explain your answer Slide 13 Hypothesis Tests About µ 1 − µ 2 When
σ 1 and σ 2 Unknown & Small Samples
Using Data Analysis in Excel
Excel’s “tTest: Two Sample Assuming Unequal Variances” Tool
s Step 1 Select the Tools menu
Step 2 Choose the Data Analysis option
Step 3 Choose tTest: Two Sample Assuming Unequal Variances from the list of Analysis Tools
… continued Slide 14 Solution to #1 Using Data Analysis in Excel
s Excel Dialog Box Slide 15 Solution to #1 Using Data Analysis in Excel H0: µ1 = µ2 H : µ # µ Slide 16 Hypothesis Tests About µ 1 − µ 2 When
σ 1 and σ 2 Unknown & Small Samples
s SWStat+ “tTest: Two Sample Assuming Unequal Variances” Tool Step 1 Create Data area
Step 2 Choose Statistics, and then choose the Intervals and Tests option
Step 3 From Type, Choose Two Samples, Different Means, Unequal Variances, tTest
… continued Slide 17 Solution to #1 Using SWStat+
(Creating Data Area) Data Area Slide 18 Solution to #1 Using SWStat+
(Selecting Intervals and Tests Option) Slide 19 Solution to #1 Using SWStat+
(Results) H0: µ1 = µ2 H : µ # µ Slide 20 Hypothesis Tests About µ 1 = µ 2 When
σ 1 and σ 2 Unknown
Challenging Example #2
s Information regarding the ACT scores of
samples of twenty two students in two
different majors at CSU is given in next slide. s Questions: (a) Use Excel, and at 95%
confidence test to determine whether there is
a significant difference in the means of the two
populations, and (b) explain your answer. Slide 21 DATA Hypothesis Test H0: µ1 = µ2 Ha: µ1 # µ2 Slide 22 Solution to #2 Using Data Analysis in Excel Slide 23 Solution to #2 Using SWStat+
(Results)
Two Samples, Different Means, Unequal Variances, t Test H0: µ1 = µ2 Ha: µ1 # µ2 Slide 24 Hypothesis Tests About µ 1 − µ 2 When
σ 1 and σ 2 are Known
s s s If samples are large (n1≥30 and n2≥30) and also if population standard deviations (σ1 and σ2) are given then you can use either Data Analysis or SWStat+ in Excel to solve the problem. Note that in this case, the option that you use in Data Analysis is zTest: Two Sample for Means and the option that you use in SWStat+ is Two Sample, diff Variances/SDs: F
See next example. Slide 25 Hypothesis Tests About µ 1 − µ 2 When
σ 1 and σ 2 are Known s Example: Par, Inc. Par, Inc. is a manufacturer
of golf equipment and has
developed a new golf ball
that has been designed to
provide “extra distance.” In a test of driving distance using a mechanical
driving device, a sample of Par golf balls was
compared with a sample of golf balls made by Rap,
Ltd., a competitor. The sample statistics appear on the
next slide. Slide 26 Hypothesis Testing of µ1 µ2: σ 1 and σ 2 Known s Example: Par, Inc. Sample Size
Sample Mean Sample #1
Sample #2
Par, Inc.
Rap, Ltd. 120 balls 80 balls
275 yards 258 yards Given that the two population standard deviations are
known as: σ 1 = 15 yards and σ 2 = 20 yards, then
known Can we conclude, using α = 0.01, that the mean
driving distance of Par, Inc. golf balls is greater than
the mean driving distance of Rap, Ltd. golf balls? Slide 27 Hypothesis Testing of µ1 µ2 When σ 1 and σ 2 are Known p –Value and Critical Value Approaches
The hypotheses is: H0: µ1 µ2 < 0 Ha: µ1 µ2 > 0 where: µ1 = mean distance for the population of Par, Inc. golf balls
µ2 = mean distance for the population of Rap, Ltd. golf balls
The level of significance is: α = .01 Slide 28 Hypothesis Tests About µ 1 − µ 2 When
σ 1 and σ 2 are Known
s Excel’s “zTest: Two Sample for Means” Tool
Step 1 After data entry, then select the Tools menu
Step 2 Choose the Data Analysis option
Step 3 Choose zTest: Two Sample for Means from the list of Analysis Tools
… continued Slide 29 Hypothesis Tests About µ 1 − µ 2 When σ 1 and σ 2 are Known s Excel Dialog Box H0: µ1 µ2 < 0 H : µ µ > 0 OR H0: µ1 ≤ µ2 H : µ > µ Slide 30 Hypothesis Tests About µ 1 − µ 2 When
σ 1 and σ 2 are Known s 1
2
3
4
5
6
7
8
9
10
11
12
13 Excel Value Worksheet
A
Par
255
270
294
245
300
262
281
257
268
295
249
291 BC
D
Rap
266
zTest: Two Sample for Means
238
243
277
Mean
275
Known Variance
244
Observations
239
Hypothesized Mean Difference
242
z
280
P(Z<=z) onetail
261
z Critical onetail
276
P(Z<=z) twotail
241
z Critical twotail Note: Rows 14121 are not shown. E F P ar, Inc.
Rap, Ltd.
235
218
225
400
120
80
0
6.483545607
4.47959E11
2.326347874
8.95919E11
2.575829304 Slide 31 Conclusion
s s Because p–value = 0 < α = 0.01 and also because actual z = 6.49 > critical z = 2.33 we reject H0. In other words: At the 0.01 level of significance, the sample evidence
indicates that the mean driving distance of Par, Inc. golf
balls ( µ1) is greater than the mean driving distance of
Rap, Ltd. golf balls (µ2). In other words
H0: µ1 µ2 < 0 Ha: µ1 µ2 > 0 OR H0: µ1 ≤ µ2 Ha: µ1 > µ2 Slide 32 End of Chapter 11 DANCER ...
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 Fall '09
 Statistics, Standard Deviation, Null hypothesis, Statistical hypothesis testing, Household income in the United States

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