Questions: After asking 7 randomly selected residents from her town whether or not they listen to country music, Lisa reports that 29 % of the residents from her town listen to country music based on her survey. Why is this statistic misleading? Select the correct answer below: The statistic results from a calculation error. The data contain one or more outliers. The sample size is insufficient. The sample is biased.

Solution Steps

To determine the required sample size for estimating the population proportion with a specified confidence level, we first calculate the Z-score corresponding to a confidence level of \( 0.95 \). The Z-score is given by:

\[ Z = \text{PPF}\left(1 - \frac{1 - 0.95}{2}\right) = \text{PPF}(0.975) = 1.96 \]

Using the Z-score, the population standard deviation \( \sigma = 0.5 \), and the desired margin of error \( E = 0.05 \), we can calculate the required sample size \( n \) using the formula:

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

Substituting the values:

\[ n = \left(\frac{1.96 \cdot 0.5}{0.05}\right)^2 = (19.6)^2 = 384.16 \approx 385 \]

Lisa's sample size is \( n_L = 7 \). We compare this with the calculated required sample size:

\[ \text{Calculated Sample Size} = 385 \] \[ \text{Lisa's Sample Size} = 7 \]

Since \( 7 < 385 \), we conclude that Lisa's sample size is insufficient.

Final Answer