Questions: a) Assume that nothing is known about the percentage of adults who have heard of the brand. n=267 (Round up to the nearest integer.) b) Assume that a recent survey suggests that about 83% of adults have heard of the brand. n=151 (Round up to the nearest integer.) c) Given that the required sample size is relatively small, could he simply survey the adults at the nearest college? adults. B. No, a sample of students at the nearest college is a cluster sample, not a simple random sample, so it is very possible that the results would not be representative of the population of adults. C. Yes, a sample of students at the nearest college is a simple random sample, so the results should be representative of the population of adults. D. No, a sample of students at the nearest college is a stratified sample, not a simple random sample, so it is very possible that the results would not be representative of the population of adults.

a) Assume that nothing is known about the percentage of adults who have heard of the brand. n=267 (Round up to the nearest integer.) b) Assume that a recent survey suggests that about 83% of adults have heard of the brand. n=151 (Round up to the nearest integer.) c) Given that the required sample size is relatively small, could he simply survey the adults at the nearest college? adults. B. No, a sample of students at the nearest college is a cluster sample, not a simple random sample, so it is very possible that the results would not be representative of the population of adults. C. Yes, a sample of students at the nearest college is a simple random sample, so the results should be representative of the population of adults. D. No, a sample of students at the nearest college is a stratified sample, not a simple random sample, so it is very possible that the results would not be representative of the population of adults.
Transcript text: a) Assume that nothing is known about the percentage of adults who have heard of the brand. \[ n=267 \] (Round up to the nearest integer.) b) Assume that a recent survey suggests that about $83 \%$ of adults have heard of the brand. \[ n=151 \] (Round up to the nearest integer.) c) Given that the required sample size is relatively small, could he simply survey the adults at the nearest college? adults. B. No, a sample of students at the nearest college is a cluster sample, not a simple random sample, so it is very possible that the results would not be representative of the population of adults. C. Yes, a sample of students at the nearest college is a simple random sample, so the results should be representative of the population of adults. D. No, a sample of students at the nearest college is a stratified sample, not a simple random sample, so it is very possible that the results would not be representative of the population of adults.
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Solution

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Solution Steps

Solution Approach

a) To determine the sample size needed when nothing is known about the percentage of adults who have heard of the brand, we use the formula for sample size calculation for proportions with a 50% estimate (p = 0.5), which maximizes the sample size. The formula is: \[ n = \left( \frac{Z^2 \cdot p \cdot (1 - p)}{E^2} \right) \] where \( Z \) is the Z-value (e.g., 1.96 for 95% confidence), \( p \) is the estimated proportion (0.5), and \( E \) is the margin of error.

b) When a recent survey suggests that about 83% of adults have heard of the brand, we use the same formula but with \( p = 0.83 \).

c) For the third part, we need to determine if surveying adults at the nearest college would be a representative sample. This involves understanding the types of sampling methods.

Step 1: Calculate Sample Size When Nothing is Known About the Percentage

To determine the sample size needed when nothing is known about the percentage of adults who have heard of the brand, we use the formula for sample size calculation for proportions with a 50% estimate (\( p = 0.5 \)), which maximizes the sample size. The formula is: \[ n = \left( \frac{Z^2 \cdot p \cdot (1 - p)}{E^2} \right) \] where \( Z = 1.96 \) (Z-value for 95% confidence), \( p = 0.5 \), and \( E = 0.05 \) (margin of error).

Plugging in the values: \[ n = \left( \frac{1.96^2 \cdot 0.5 \cdot (1 - 0.5)}{0.05^2} \right) \approx 384.16 \]

Rounding up to the nearest integer: \[ n = 385 \]

Step 2: Calculate Sample Size When 83% of Adults Have Heard of the Brand

When a recent survey suggests that about 83% of adults have heard of the brand, we use the same formula but with \( p = 0.83 \).

\[ n = \left( \frac{1.96^2 \cdot 0.83 \cdot (1 - 0.83)}{0.05^2} \right) \approx 216.68 \]

Rounding up to the nearest integer: \[ n = 217 \]

Step 3: Determine if Surveying Adults at the Nearest College is Representative

For the third part, we need to determine if surveying adults at the nearest college would be a representative sample. The answer is: \[ \text{B. No, a sample of students at the nearest college is a cluster sample, not a simple random sample, so it is very possible that the results would not be representative of the population of adults.} \]

Final Answer

\[ \boxed{ \begin{array}{l} \text{a) Sample size needed when nothing is known about the percentage: } 385 \\ \text{b) Sample size needed when 83% of adults have heard of the brand: } 217 \\ \text{c) The answer is B. No, a sample of students at the nearest college is a cluster sample, not a simple random sample, so it is very possible that the results would not be representative of the population of adults.} \end{array} } \]

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