Homework 003
Homework 3
1.
M&M/MARS, maker of M&M Chocolate candies, conducted a national poll in which more than 10 million people indicated their preference for a new color. The tally of this poll resulted in the replacement of tan-colored M&Ms with a new blue color. In the brochure “Colors,” made available by M&M/MARS Consumer Affairs, the distribution of colors of the plain color is as follows:
Brown
Yellow
Red
Orange
Green
Blue
30%
20%
20%
10%
10%
10%
In a follow-up study, samples of 1-pound bags were used to determine whether the reported percentages were indeed valid. The following results were obtained for one sample of 506 plain candies.
Brown
Yellow
Red
Orange
Green
Blue
177
135
79
41
36
38
Use α=0.05 to determine whether these data support the percentages reported by the company.
Information
??? = ??????????? ?? ????? ???????
??? = ??????????? ?? ?????? ???????
??? = ??????????? ?? ??? ???????
??? = ??????????? ?? ?????? ???????
??? = ??????????? ?? ????? ???????
??? = ??????????? ?? ???? ???????
Sol
????????
?????????
(?? )
?????
????????????
??????????
?????
30%
177.0
??????
20%
135.0
???
20%
79.0
??????
10%
41.0
?????
10%
36.0
????
10%
38.0
?????
???. ?
????????
?????????
(?? )
506 ∗ 30
= 151.8
100
506 ∗ 20
= 101.2
100
506 ∗ 20
= 101.2
100
506 ∗ 10
= 50.6
100
506 ∗ 10
= 50.6
100
506 ∗ 10
= 50.6
100
(?? − ?? )
(?? − ?? )?
(?? − ?? )?
??
25.2
635.04
4.1834
33.8
1142.44
11.2889
−22.2
492.84
4.8700
−9.6
92.16
1.8213
−14.6
213.16
4.2126
−12.6
158.76
3.1375
???. ?
??. ????
1) ?0 : ??? = 0.3, ??? = 0.2, ??? = 0.2, ??? = 0.1, ???? = 0.1, ??? = 0.1
?1 : ??? ≠ 0.3, ??? ≠ 0.2, ??? ≠ 0.2, ??? ≠ 0.1, ??? ≠ 0.1, ??? ≠ 0.1
2) ∝ = 0.05
6
3) ??2 = ∑
?=1
(?? − ?? )2
= 29.5137
??
4) ?? = 5, ? − ????? = 1.8382 × 10−5
5) ??????? ? − ????? ???ℎ ∝
-
?? ? − ????? < ∝, ?????? ?0
-
??ℎ????, ?????? ?0
6) 1.8382 × 10−5 < 0.05, ?????? ?0
Therefore, we find sufficient statistical evidence to reject the null
1.
M&M/MARS, maker of M&M Chocolate candies, conducted a national poll in which more than 10 million people indicated their preference for a new color. The tally of this poll resulted in the replacement of tan-colored M&Ms with a new blue color. In the brochure “Colors,” made available by M&M/MARS Consumer Affairs, the distribution of colors of the plain color is as follows:
Brown
Yellow
Red
Orange
Green
Blue
30%
20%
20%
10%
10%
10%
In a follow-up study, samples of 1-pound bags were used to determine whether the reported percentages were indeed valid. The following results were obtained for one sample of 506 plain candies.
Brown
Yellow
Red
Orange
Green
Blue
177
135
79
41
36
38
Use α=0.05 to determine whether these data support the percentages reported by the company.
Information
??? = ??????????? ?? ????? ???????
??? = ??????????? ?? ?????? ???????
??? = ??????????? ?? ??? ???????
??? = ??????????? ?? ?????? ???????
??? = ??????????? ?? ????? ???????
??? = ??????????? ?? ???? ???????
Sol
????????
?????????
(?? )
?????
????????????
??????????
?????
30%
177.0
??????
20%
135.0
???
20%
79.0
??????
10%
41.0
?????
10%
36.0
????
10%
38.0
?????
???. ?
????????
?????????
(?? )
506 ∗ 30
= 151.8
100
506 ∗ 20
= 101.2
100
506 ∗ 20
= 101.2
100
506 ∗ 10
= 50.6
100
506 ∗ 10
= 50.6
100
506 ∗ 10
= 50.6
100
(?? − ?? )
(?? − ?? )?
(?? − ?? )?
??
25.2
635.04
4.1834
33.8
1142.44
11.2889
−22.2
492.84
4.8700
−9.6
92.16
1.8213
−14.6
213.16
4.2126
−12.6
158.76
3.1375
???. ?
??. ????
1) ?0 : ??? = 0.3, ??? = 0.2, ??? = 0.2, ??? = 0.1, ???? = 0.1, ??? = 0.1
?1 : ??? ≠ 0.3, ??? ≠ 0.2, ??? ≠ 0.2, ??? ≠ 0.1, ??? ≠ 0.1, ??? ≠ 0.1
2) ∝ = 0.05
6
3) ??2 = ∑
?=1
(?? − ?? )2
= 29.5137
??
4) ?? = 5, ? − ????? = 1.8382 × 10−5
5) ??????? ? − ????? ???ℎ ∝
-
?? ? − ????? < ∝, ?????? ?0
-
??ℎ????, ?????? ?0
6) 1.8382 × 10−5 < 0.05, ?????? ?0
Therefore, we find sufficient statistical evidence to reject the null