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.
failed

Solution

failed
failed

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

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

Was this solution helpful?
failed
Unhelpful
failed
Helpful

Questions asked by other users

failedAnswered
Calculate the amount of heat needed to melt 176 g of ice (H2O) and bring it to a temperature of 90.4°C. Be sure your answer has a unit symbol and the correct number of significant digits.
failedAnswered
True or false? A patient calls complaining of painful urination with traces of blood in the urine. The patient's physician is out for the day. The medical assistant should schedule an appointment for the next day. True False
failedAnswered
Question 1 2 pts Which of the following statements are critical to understanding the impact of diversity on early childhood classrooms? (RW, pg. 10) The United States is becoming less diverse, which is reflected in current education standards. Most education in the United States effectively addresses the needs of students of color by considering diversity. US standards in early childhood education adequately address diversity and equity. Teachers and administrators often lack training or mentorship on how to implement culturally relevant anti-bias education.
failedAnswered
An analytical chemist weighs out 0.165 g of an unknown diprotic acid into a 250 mL volumetric flask and dilutes to the mark with distilled water. He then titrates this solution with 0.0600 M NaOH solution. When the titration reaches the equivalence point, the chemist finds he has added 41.0 mL of NaOH solution. Calculate the molar mass of the unknown acid. Be sure your answer has the correct number of significant digits.
failedAnswered
Liquid nitrogen, N2(l), boils at -195.8°C. What is the boiling point of liquid nitrogen in Kelvin? Express your answer to one place past the decimal. If your answer is negative, include the sign.