Solutions of QUIZ 7. M262 Section 1. (11.05.2022).
1. An automobile manufacturer is interested in determining the relationship between the
size and manufacturer of newly purchased automobiles. One thousand recent buyers of
American-made cars are randomly sampled, and each purchase is classified with respect to
the size and manufacturer of the automobile.
Out of 341 cars of manufacturer A 157 are small, 126 are intermediate, 58 are large.
Out of 192 cars of manufacturer B 65 are small, 82 are intermediate, 45 are large.
Out of 383 cars of manufacturer C 181 are small, 142 are intermediate, 60 are large.
Out of 84 cars of manufacturer D 10 are small, 46 are intermediate, 28 are large.
At α = 0.01 test to determine if the two classifications, manufacturer and size, are
dependent.
Solution:
H0 : Manufacturer and size are not dependent.
H1 : Manufacturer and size are dependent.
Let us calculate expected cell frequencies under the assumption that the classifications are
independent. There are 413 Small, 396 Intermediate, 191 Large automobiles.
413 · 341 396 · 341
Therefore, for 157 we expect = 140.833, for 126 we expect = 135.036,
1000 1000
191 · 341
for 58 we expect = 65.131.
1000
413 · 192 396 · 192
For 65 we expect = 79.296, for 82 we expect = 135.036, for 45 we
1000 1000
191 · 192
expect = 36.672.
1000
413 · 383 396 · 383
For 181 we expect = 158.179, for 142 we expect = 151.668, for 60 we
1000 1000
191 · 383
expect = 73.153.
1000
413 · 84 396 · 84
For 10 we expect = 34.692, for 46 we expect = 33.264, for 28 we expect
1000 1000
191 · 84
= 36.672.
1000
Rejection region: degree of freedom is (3 − 1)(4 − 1) = 6. χ20.01 = 16.81. Test statistics:
(Oi − Ei )2
χ2 =
X
= 45.81
Ei
Since χ2 = 45.81 > 16.81 = χ20.01 we reject H0 and accept H1 : we conclude that
manufacturer and size are dependent..
This study source was downloaded by 100000899606396 from CourseHero.com on 09-25-2025 13:14:52 GMT -05:00
1
https://www.coursehero.com/file/249824214/22262q7solpdf/
1. An automobile manufacturer is interested in determining the relationship between the
size and manufacturer of newly purchased automobiles. One thousand recent buyers of
American-made cars are randomly sampled, and each purchase is classified with respect to
the size and manufacturer of the automobile.
Out of 341 cars of manufacturer A 157 are small, 126 are intermediate, 58 are large.
Out of 192 cars of manufacturer B 65 are small, 82 are intermediate, 45 are large.
Out of 383 cars of manufacturer C 181 are small, 142 are intermediate, 60 are large.
Out of 84 cars of manufacturer D 10 are small, 46 are intermediate, 28 are large.
At α = 0.01 test to determine if the two classifications, manufacturer and size, are
dependent.
Solution:
H0 : Manufacturer and size are not dependent.
H1 : Manufacturer and size are dependent.
Let us calculate expected cell frequencies under the assumption that the classifications are
independent. There are 413 Small, 396 Intermediate, 191 Large automobiles.
413 · 341 396 · 341
Therefore, for 157 we expect = 140.833, for 126 we expect = 135.036,
1000 1000
191 · 341
for 58 we expect = 65.131.
1000
413 · 192 396 · 192
For 65 we expect = 79.296, for 82 we expect = 135.036, for 45 we
1000 1000
191 · 192
expect = 36.672.
1000
413 · 383 396 · 383
For 181 we expect = 158.179, for 142 we expect = 151.668, for 60 we
1000 1000
191 · 383
expect = 73.153.
1000
413 · 84 396 · 84
For 10 we expect = 34.692, for 46 we expect = 33.264, for 28 we expect
1000 1000
191 · 84
= 36.672.
1000
Rejection region: degree of freedom is (3 − 1)(4 − 1) = 6. χ20.01 = 16.81. Test statistics:
(Oi − Ei )2
χ2 =
X
= 45.81
Ei
Since χ2 = 45.81 > 16.81 = χ20.01 we reject H0 and accept H1 : we conclude that
manufacturer and size are dependent..
This study source was downloaded by 100000899606396 from CourseHero.com on 09-25-2025 13:14:52 GMT -05:00
1
https://www.coursehero.com/file/249824214/22262q7solpdf/
