Question

2. Kelloggs claims the six fruit flavors in its Fruit Loop cereal distribute uniformly. You randomly collect 120 Fruit Loops and count the amount of each flavor. The results are below. At a 1% level of significance, should we accept their claim that the flavors distribute uniformly? (10pts) Flavors: Orange Lemon Cherry Raspberry Blueberry Lime Count: 28 21 16 25 14 16

Answer 1

Flavors	Count ( O )	Expected frequency E		(O; - E;)	(O; - E;) Ei
Orange	28	20	8	64	3.2
Lemon	21	20	1	1	0.05
Cherry	16	20	-4	16	0.8
Raspberry	25	20	5	25	1.25
blueberry	14	20	-6	36	1.8
Lime	16	20	4	16	0.8
Total	120	120			7.9

Because six fruit flavors are expected to distribute uniformly,

$Expected\, frequency =E_i=\frac{120 }{6}=20$

Null Hypothesis: : the flavors are uniformly distributed.

он

Alternative hypothesis: $H_1$ : the flavors are not uniformly distributed.

Test statistic:

$\chi ^2=\sum_{i=1}^6\left ( \frac{(O_i-E_i)^2}{E_i} \right )=7.9$

$\therefore \chi^2_{calculated}=7.9$

Level of significance=$\alpha =1\%=0.01$

dof= $\nu=n-1=6-1=5.$

Critical value:

$\\\chi^2_{\nu}(\alpha )=\chi^2_{5}(0.01 )=14.684....(from\; tables) \\\\\chi^2_{tabulated}=14.684$

Decision:

$7.9<14.684 \Rightarrow \chi ^2_{calculated}< \chi ^2_{tabulated}$

Thus we accept .

он

Conclusion: There is enough evidence at 1% l.o.s to accept the claim that flavors are uniformly distributed.