Questions: The brand manager for a brand of toothpaste must plan a campaign designed to increase brand recognition. He wants to first determine the percentage of adults who have heard of the brand. How many adults must he survey in order to be 90% confident that his estimate is within eight percentage points of the true population percentage? Complete parts (a) through (c) below. n=106 (Round up to the nearest integer.) b) Assume that a recent survey suggests that about 79% of adults have heard of the brand. n= (Round up to the nearest integer.)

Transcript text: The brand manager for a brand of toothpaste must plan a campaign designed to increase brand recognition. He wants to first determine the percentage of adults who have heard of the brand. How many adults must he survey in order to be $90 \%$ confident that his estimate is within eight percentage points of the true population percentage? Complete parts (a) through (c) below. \[ n=106 \] (Round up to the nearest integer.) b) Assume that a recent survey suggests that about $79 \%$ of adults have heard of the brand. $\mathrm{n}=\square$ $\square$ (Round up to the nearest integer.)

Solution

Solution Steps

To determine the sample size needed for a survey with a given confidence level and margin of error, we can use the formula for sample size in proportion estimation. The formula is:

\[ n = \left( \frac{Z^2 \cdot p \cdot (1 - p)}{E^2} \right) \]

where:

$ Z $ is the Z-score corresponding to the desired confidence level.
$ p $ is the estimated proportion of the population.
$ E $ is the margin of error.

For part (b), we are given:

Confidence level: 90% (Z-score for 90% confidence is approximately 1.645)
Estimated proportion ($ p $): 0.79
Margin of error ($ E $): 0.08

We will plug these values into the formula to calculate the required sample size.

Step 1: Identify Given Values

We are given the following values:

Confidence level: $90\%$
Z-score for $90\%$ confidence: $1.645$
Estimated proportion ($p$): $0.79$
Margin of error ($E$): $0.08$

Step 2: Apply the Sample Size Formula

The formula for calculating the sample size $n$ is: \[ n = \left( \frac{Z^2 \cdot p \cdot (1 - p)}{E^2} \right) \]

Step 3: Substitute the Given Values

Substitute the given values into the formula: \[ n = \left( \frac{1.645^2 \cdot 0.79 \cdot (1 - 0.79)}{0.08^2} \right) \]

Step 4: Perform the Calculation

Calculate the value step-by-step: \[ n = \left( \frac{1.645^2 \cdot 0.79 \cdot 0.21}{0.08^2} \right) \] \[ n = \left( \frac{2.706025 \cdot 0.79 \cdot 0.21}{0.0064} \right) \] \[ n = \left( \frac{0.4491}{0.0064} \right) \] \[ n \approx 70.1452 \]

Step 5: Round Up to the Nearest Integer

Since the sample size must be an integer, we round up: \[ n \approx 71 \]