Q. 1 - The Practice of Statistics for AP

Question

A company claims that each batch of its deluxe mixed nuts contains $52\%$ cashews, $27\%$ almonds, $13\%$ macadamia nuts, and $8\%$ brazil nuts. To test this claim, a quality control inspector takes a random sample of $150$ nuts from the latest batch. The one-way table below displays the sample data.

(a) State appropriate hypotheses for performing a test of the company’s claim. 

(b) Calculate the expected counts for each type of nut. Show your work 

Step-by-Step Solution

Answer

(a)$H_0:p_c=52\%,p_a=27\%,p_m=13\%,p_b=8\%$; $H_1$ : One of the $Pi$ test is wrong.

(b) The expected count of nuts are : 

Cashews	Almonds	Macadamia	Brazil
78	40.5	19.5	12

Part (a) Step 1: Given Information

Given in the question that, the each batch of mixed nuts contains $52\%$ cashews, $27\%$ almonds, $13\%$ macadamia nuts, and $8\%$ brazil nuts.

Part (a) Step 2: Explanation

$H_0$: will be the null hypothesis. The observed count of each type of nut in a group of $150$ nuts from the most recent batch matches the reported count of nuts in a group of $150$

$H_0:p_c=52\%,p_a=27\%,p_m=13\%,p_b=8\%$

$H_1$: The observed counts of each type of nut in a group of  $150$ nuts from the most recent batch do not match the reported count of nuts in a $150$ group.

Because the claim specifies a proportion of each type of nut of $100$, we can calculate the proportion of each type of nut in 150 by multiplying each number by $150.$

Part (b) Step 1: Given Information

Given in the question that,

Part (b) Step 2: Explanation

According to the information, 

Here,

$n=150$

$P\left(\text{ cashews }\right)=52\%=0.52$

$P\left(\text{ almonds }\right)=27\%=0.27$

$P\left(\text{ Macadamia }\right)=13\%=0.13$

$P\left(brazil\right)=8\%=0.08$

The expected count is then calculated as the product of the sample size $n$ and the type probability:

Cashews:

$150(0.52)=78$

Almonds:

$150(0.27)=40.5$

Macadamia:

$150(0.13)=19.5$

Brazil:

$150(0.08)=12$

Therefore, the expected counts of nuts are

$\begin{array}{|cccc|}\hline \text{ Cashews }& \text{ Almonds }& \text{ Macadamia }& \text{ Brazil }\\ 78& 40.5& 19.5& 12\\ \hline\end{array}$