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Do families and singles have the same distribution of cars? Use a level of significance of 0.05. Suppose that 100 randomly selected families and 200 randomly selected singles were asked what type of car they drove: sport, sedan, hatchback, truck, van/SUV. The results are shown in [link] . Do families and singles have the same distribution of cars? Test at a level of significance of 0.05.
Sport | Sedan | Hatchback | Truck | Van/SUV | |
---|---|---|---|---|---|
Family | 5 | 15 | 35 | 17 | 28 |
Single | 45 | 65 | 37 | 46 | 7 |
With a p -value of almost zero, we reject the null hypothesis. The data show that the distribution of cars is not the same for families and singles.
Both before and after a recent earthquake, surveys were conducted asking voters which of the three candidates they planned on voting for in the upcoming city council election. Has there been a change since the earthquake? Use a level of significance of 0.05. [link] shows the results of the survey. Has there been a change in the distribution of voter preferences since the earthquake?
Perez | Chung | Stevens | |
Before | 167 | 128 | 135 |
After | 214 | 197 | 225 |
H
_{0} : The distribution of voter preferences was the same before and after the earthquake.
H
_{a} : The distribution of voter preferences was not the same before and after the earthquake.
Degrees of Freedom (
df ):
df = number of columns – 1 = 3 – 1 = 2
Distribution for the test:
${\chi}_{2}^{2}$
Calculate the test statistic :
χ
^{2} = 3.2603 (calculator or computer)
Probability statement:
p -value=
P (
χ
^{2} >3.2603) = 0.1959
Press the
MATRX
key and arrow over to
EDIT
. Press
1:[A]
. Press
2 ENTER 3 ENTER
. Enter the table values by row. Press
ENTER
after each. Press
2nd QUIT
. Press
STAT
and arrow over to
TESTS
. Arrow down to
C:χ2-TEST
. Press
ENTER
. You should see
Observed:[A] and Expected:[B]
. Arrow down to
Calculate
. Press
ENTER
. The test statistic is 3.2603 and the
p -value = 0.1959. Do the procedure a second time but arrow down to
Draw
instead of
calculate
.
Compare α and the p -value: α = 0.05 and the p -value = 0.1959. α < p -value.
Make a decision: Since α < p -value, do not reject H _{o} .
Conclusion: At a 5% level of significance, from the data, there is insufficient evidence to conclude that the distribution of voter preferences was not the same before and after the earthquake.
Ivy League schools receive many applications, but only some can be accepted. At the schools listed in [link] , two types of applications are accepted: regular and early decision.
Application Type Accepted | Brown | Columbia | Cornell | Dartmouth | Penn | Yale |
---|---|---|---|---|---|---|
Regular | 2,115 | 1,792 | 5,306 | 1,734 | 2,685 | 1,245 |
Early Decision | 577 | 627 | 1,228 | 444 | 1,195 | 761 |
We want to know if the number of regular applications accepted follows the same distribution as the number of early applications accepted. State the null and alternative hypotheses, the degrees of freedom and the test statistic, sketch the graph of the p -value, and draw a conclusion about the test of homogeneity.
H _{0} : The distribution of regular applications accepted is the same as the distribution of early applications accepted.
H
_{a} : The distribution of regular applications accepted is not the same as the distribution of early applications accepted.
df = 5
χ
^{2} test statistic = 430.06
Press the
MATRX
key and arrow over to
EDIT
. Press
1:[A]
. Press
3 ENTER 3 ENTER
. Enter the table values by row. Press
ENTER
after each. Press
2nd QUIT
. Press
STAT
and arrow over to
TESTS
. Arrow down to
C:χ2-TEST
. Press
ENTER
. You should see
Observed:[A] and Expected:[B]
. Arrow down to
Calculate
. Press
ENTER
. The test statistic is 430.06 and the
p -value = 9.80E-91. Do the procedure a second time but arrow down to
Draw
instead of
calculate
.
Data from the Insurance Institute for Highway Safety, 2013. Available online at www.iihs.org/iihs/ratings (accessed May 24, 2013).
“Energy use (kg of oil equivalent per capita).” The World Bank, 2013. Available online at http://data.worldbank.org/indicator/EG.USE.PCAP.KG.OE/countries (accessed May 24, 2013).
“Parent and Family Involvement Survey of 2007 National Household Education Survey Program (NHES),” U.S. Department of Education, National Center for Education Statistics. Available online at http://nces.ed.gov/pubsearch/pubsinfo.asp?pubid=2009030 (accessed May 24, 2013).
“Parent and Family Involvement Survey of 2007 National Household Education Survey Program (NHES),” U.S. Department of Education, National Center for Education Statistics. Available online at http://nces.ed.gov/pubs2009/2009030_sup.pdf (accessed May 24, 2013).
To assess whether two data sets are derived from the same distribution—which need not be known, you can apply the test for homogeneity that uses the chi-square distribution. The null hypothesis for this test states that the populations of the two data sets come from the same distribution. The test compares the observed values against the expected values if the two populations followed the same distribution. The test is right-tailed. Each observation or cell category must have an expected value of at least five.
$\sum}_{i\cdot j}\frac{{(O-E)}^{2}}{E$ Homogeneity test statistic where:
O = observed values
E = expected values
i = number of rows in data contingency table
j = number of columns in data contingency table
df = ( i −1)( j −1) Degrees of freedom
A math teacher wants to see if two of her classes have the same distribution of test scores. What test should she use?
test for homogeneity
What are the null and alternative hypotheses for [link] ?
A market researcher wants to see if two different stores have the same distribution of sales throughout the year. What type of test should he use?
test for homogeneity
A meteorologist wants to know if East and West Australia have the same distribution of storms. What type of test should she use?
What condition must be met to use the test for homogeneity?
All values in the table must be greater than or equal to five.
Use the following information to answer the next five exercises: Do private practice doctors and hospital doctors have the same distribution of working hours? Suppose that a sample of 100 private practice doctors and 150 hospital doctors are selected at random and asked about the number of hours a week they work. The results are shown in [link] .
20–30 | 30–40 | 40–50 | 50–60 | |
---|---|---|---|---|
Private Practice | 16 | 40 | 38 | 6 |
Hospital | 8 | 44 | 59 | 39 |
State the null and alternative hypotheses.
What is the test statistic?
What can you conclude at the 5% significance level?
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