Chapter 11: Q.16 (page 694)

Orange, lemon, cherry, raspberry, blueberry, and lime are among the six fruit flavours available in Kellogg's Froot Loops cereal. Charise counted the number of cereal pieces in each flavour as she poured out her morning bowl of cereal. Here are her statistics.
Test the null hypothesis that each flavour of Kellogg's Froot Loops is distributed evenly throughout the population. Perform a follow-up analysis if you discover a noteworthy result.

Short Answer

Expert verified

There is insufficient data to establish that the produced population of Froot Loops contains an equal amount of each taste.

Step by step solution

Given information

Given in the question that, Kellogg’s Froot Loops cereal comes in six fruit flavors: orange, lemon, cherry, raspberry, blueberry, and lime. Charise poured out her morning bowl of cereal and methodically counted the number of cereal pieces of each flavor. Here are her data:

Explanation

Given,

Flavour	Count
Orange	$28$
Lemon	$21$
Cherry	$16$
Rasberry	$25$
Blueberry	$14$
Lime	$16$

The test statistic will be calculated using the formula below.

$χ^{2} = \sum \frac{(O - E)^{2}}{E}$

The following are the null and alternative hypotheses:

localid="1650619689827" $H_{0} : p_{1} = \frac{1}{6}$

$H_{0} : p_{2} = \frac{1}{6}$

$H_{0} : p_{3} = \frac{1}{6}$

$H_{0} : p_{4} = \frac{1}{6}$

$H_{0} : p_{5} = \frac{1}{6}$

$H_{0} : p_{6} = \frac{1}{6}$

$H_{a} :$ At least one of $p_{i}$ is different

calculation for test statistic

The test statistic is calculated as follows:

Observed value	Expected value	(O-E)	${(O - E)}^{2}$	$\frac{{(O - E)}^{2}}{E}$
$28$	$20$	$8$	$64$	$3.2$
$21$	$20$	$1$	$1$	$0.05$
$16$	$20$	$- 4$	$16$	$0.8$
$25$	$20$	$5$	$25$	$1.25$
$14$	$20$	$- 6$	$36$	$1.8$
$16$	$20$	$- 4$	$16$	$0.8$
				$\sum {\frac{(O - E)^{2}}{E}}^{2} = 7.9$