1 - The Practice of Statistics for AP

Question

Aw, nuts! 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)  The null hypothesis will be $H_0$ : The observed count of each type of nuts in a group of $150$nuts from latest batch are same as claimed count of nuts in a group of $150$

Versus

$H_1$ : The observed counts of each type of nuts in a group of $150$ nuts from latest batch are not same as claimed count of nuts in a group of$150$.

(b) 

Step 1: State the Hypotheses

(a) \(H_0\): The observed proportions match the claimed proportions (52% cashews, 27% almonds, 13% macadamia, 8% brazil).
\(H_a\): At least one proportion differs from the claimed value.

Step 2: Identify the Test

This is a chi-square goodness-of-fit test. Calculate the test statistic:
\(\chi^2 = \sum \frac{(O_i - E_i)^2}{E_i}\)
where \(O_i\) are observed counts and \(E_i\) are expected counts based on the claimed proportions.

Step 3: Decision Rule

Compare \(\chi^2\) to the critical value from the chi-square distribution with \(df = 4 - 1 = 3\) degrees of freedom at the chosen significance level.