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Statistics

Imagine you are a product manager at Chips Amor Cookie Company and you want to test how accurate the claim is that your cookies have more chocolate chips than the those produced by a local grocery store brand.

You gather a team of consumers to compare the cookies. You give each participant a Chips Amor cookie in a bag labeled A and a local grocery store brand cookie in a bag labeled B. They are asked to count the number chips in each cookie. You have 30 participants.

What parameters would they be comparing?
How can you write a null hypothesis and an alternative hypothesis?
What are the populations from which the samples came?
Based on your hypothesis, is this a one-tailed or two-tailed test?

Write a null hypothesis and a research hypothesis:

So, are the samples of cookies random?
Are the two samples independent of each other?

Full Answer Section

- In mathematical terms: $μ_{C hi p s A m or} \leq μ_{L oc a lB r an d}$
- Alternatively, a more common null for a one-tailed test: $μ_{C hi p s A m or} = μ_{L oc a lB r an d}$
Alternative Hypothesis ( $H_{1}$ or $H_{a}$ ): Chips Amor cookies have a significantly greater average number of chocolate chips than the local grocery store brand cookies.
- In mathematical terms: $μ_{C hi p s A m or} > μ_{L oc a lB r an d}$

Populations from Which the Samples Came

Population 1: All Chips Amor cookies ever produced (or that could be produced under the current manufacturing process).
Population 2: All cookies produced by the specific local grocery store brand (or that could be produced under their current manufacturing process).

Your 30 Chips Amor cookies are a sample from Population 1, and your 30 local brand cookies are a sample from Population 2.

One-tailed or Two-tailed Test?

Based on your hypothesis, this is a one-tailed test.

Explanation: You are specifically testing for a difference in one direction – whether Chips Amor cookies have more chips. You are not interested in whether they simply have a different number (either more or fewer), but explicitly more. If your alternative hypothesis had been $μ_{C hi p s A m or} \neq = μ_{L oc a lB r an d}$ (i.e., just "there's a difference"), it would be a two-tailed test.

Revised Null and Research (Alternative) Hypothesis:

Let

μ_{A}

be the true average number of chocolate chips in Chips Amor cookies, and

μ_{B}

be the true average number of chocolate chips in the local grocery store brand cookies.

Null Hypothesis ( $H_{0}$ ): $μ_{A} \leq μ_{B}$ (Chips Amor cookies have the same or fewer chocolate chips than the local grocery store brand cookies).
Research Hypothesis ( $H_{1}$ ): $μ_{A} > μ_{B}$ (Chips Amor cookies have more chocolate chips than the local grocery store brand cookies).

Are the samples of cookies random?

Likely No, not truly random in a strict statistical sense, but it could be a "convenience sample" aiming for representativeness.

Why likely not truly random: A truly random sample would require:
- Access to the entire production run of Chips Amor cookies and the local brand cookies.
- A method for selecting individual cookies from that entire population in such a way that every cookie has an equal chance of being selected (e.g., pulling random bags from different production batches across different dates and locations).
What it probably is: You likely bought a few bags of each from local stores. While this is practical, it introduces potential biases. For example, the cookies might only represent specific production batches, storage conditions, or retail locations, rather than the entire universe of cookies.
How to improve randomness: To make it more random, you'd ideally:
- Source cookies from various retail locations (different grocery stores, different parts of the city/region).
- Ensure cookies come from different production dates/batches if possible.
- Use a systematic selection method once you have the bags (e.g., selecting every Nth cookie from the bags).

Sample Answer

Parameters Being Compared

The primary parameter being compared is the average number of chocolate chips per cookie between the two brands. Specifically, the participants would be comparing:

Quantitative Count: The exact numerical count of chocolate chips in each cookie they receive.
Perceived "Moreness": While the core is a quantitative count, the underlying goal is to validate a claim of "more," which implies a difference in the mean chip count.

Null Hypothesis and Alternative Hypothesis

Given the claim is "Chips Amor cookies have more chocolate chips than the local grocery store brand," this dictates the direction of your hypothesis.

Null Hypothesis ( $H_{0}$ ): There is no significant difference in the average number of chocolate chips between Chips Amor cookies and the local grocery store brand cookies, or Chips Amor cookies have fewer or the same number of chocolate chips as the local grocery store brand cookies.