Answered: The brand manager for a brand of…

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

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$### Case Study: Determining Sample Size for Brand Recognition Survey **Objective:** The brand manager for a toothpaste company aims to design a campaign to boost brand recognition. The first step involves determining the percentage of adults aware of the brand. The question is: how many adults need to be surveyed to ensure the estimate is within eight percentage points of the true population percentage, with 80% confidence? #### Problem Breakdown: **a)** **Unknown Percentage:** To calculate the required sample size when no initial data is available on the percentage of adults aware of the brand. The problem prompt includes the following details and steps for calculation: - **Required Formula:** \[ n = \left(\dfrac{Z^2 \cdot p \cdot (1-p)}{E^2}\right) \] - **Given Values:** - Confidence Level: 80%, which corresponds to $ Z \approx 1.28 $ (Z-score for 80% confidence) - Error Margin ($ E $): 8 percentage points or 0.08 Since the percentage $ p $ is not known, we use $ p = 0.5 $ for maximum variability. - **Calculation Details:** Plug in the values: \[ n = \dfrac{(1.28)^2 \cdot 0.5 \cdot (1-0.5)}{0.08^2} \] Simplify: \[ n = \dfrac{1.6384 \cdot 0.25}{0.0064} \] \[ n = \dfrac{0.4096}{0.0064} \] \[ n \approx 64 \] Thus, the brand manager needs to survey approximately 64 adults to achieve the desired confidence level and margin of error. **User Interface Elements:** - **Input Box:** Where users can enter their calculated value for $ n $. - **Buttons:** - "Check Answer" to verify the entered value. - "Clear All" to reset the input fields. - **Progress Bar:** Indicates that there are "2 parts remaining" in this section of the problem, suggesting multiple steps or components in the assignment. --- **Instructions for Students:** 1. Calculate the sample size using the provided confidence level and margin of error. 2. Input your answer in the designated box and click "Check Answer" to validate. 3.$

Transcribed Image Text:### Case Study: Determining Sample Size for Brand Recognition Survey **Objective:** The brand manager for a toothpaste company aims to design a campaign to boost brand recognition. The first step involves determining the percentage of adults aware of the brand. The question is: how many adults need to be surveyed to ensure the estimate is within eight percentage points of the true population percentage, with 80% confidence? #### Problem Breakdown: **a)** **Unknown Percentage:** To calculate the required sample size when no initial data is available on the percentage of adults aware of the brand. The problem prompt includes the following details and steps for calculation: - **Required Formula:** \[ n = \left(\dfrac{Z^2 \cdot p \cdot (1-p)}{E^2}\right) \] - **Given Values:** - Confidence Level: 80%, which corresponds to $ Z \approx 1.28 $ (Z-score for 80% confidence) - Error Margin ($ E $): 8 percentage points or 0.08 Since the percentage $ p $ is not known, we use $ p = 0.5 $ for maximum variability. - **Calculation Details:** Plug in the values: \[ n = \dfrac{(1.28)^2 \cdot 0.5 \cdot (1-0.5)}{0.08^2} \] Simplify: \[ n = \dfrac{1.6384 \cdot 0.25}{0.0064} \] \[ n = \dfrac{0.4096}{0.0064} \] \[ n \approx 64 \] Thus, the brand manager needs to survey approximately 64 adults to achieve the desired confidence level and margin of error. **User Interface Elements:** - **Input Box:** Where users can enter their calculated value for $ n $. - **Buttons:** - "Check Answer" to verify the entered value. - "Clear All" to reset the input fields. - **Progress Bar:** Indicates that there are "2 parts remaining" in this section of the problem, suggesting multiple steps or components in the assignment. --- **Instructions for Students:** 1. Calculate the sample size using the provided confidence level and margin of error. 2. Input your answer in the designated box and click "Check Answer" to validate. 3.

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