In the United States. 8% of households own a motorcycle. You plan to send surveys to an SRS of 500 households Let hat (p) be the proportion of households in the sample who own a motorcycle. (a) Describe the shape center, and variability of the sampling distribution of hat (p) Shape: square because np=square and n(1-p)=square are square Center: mu _(hat (p))=square Variability: sigma _(hat (p))=square (b) Find the probability that at least 10% of the households in the sample own a motorcycle. square (Round to 4 decimal places. Leave your answer in decimal form.)

To solve this problem, we need to analyze the sampling distribution of \(\hat{p}\), the sample proportion of households that own a motorcycle.<br /><br />### (a) Describe the shape, center, and variability of the sampling distribution of \(\hat{p}\).<br /><br />1. **Shape**: The shape of the sampling distribution of \(\hat{p}\) is approximately normal if both \(np\) and \(n(1-p)\) are greater than 10. <br /><br /> Given:<br /> - \(n = 500\)<br /> - \(p = 0.08\)<br /><br /> Calculate:<br /> \[<br /> np = 500 \times 0.08 = 40<br /> \]<br /> \[<br /> n(1-p) = 500 \times (1 - 0.08) = 500 \times 0.92 = 460<br /> \]<br /><br /> Since both \(np = 40\) and \(n(1-p) = 460\) are greater than 10, the shape of the sampling distribution is approximately **normal**.<br /><br />2. **Center**: The mean of the sampling distribution of \(\hat{p}\) is equal to the population proportion \(p\).<br /> \[<br /> \mu_{\hat{p}} = p = 0.08<br /> \]<br /><br />3. **Variability**: The standard deviation of the sampling distribution of \(\hat{p}\) is given by:<br /> \[<br /> \sigma_{\hat{p}} = \sqrt{\frac{p(1-p)}{n}} = \sqrt{\frac{0.08 \times 0.92}{500}}<br /> \]<br /> \[<br /> \sigma_{\hat{p}} = \sqrt{\frac{0.0736}{500}} = \sqrt{0.0001472} \approx 0.01213<br /> \]<br /><br />### (b) Find the probability that at least \(10\%\) of the households in the sample own a motorcycle.<br /><br />We want to find \(P(\hat{p} \geq 0.10)\).<br /><br />First, calculate the z-score for \(\hat{p} = 0.10\):<br />\[<br />z = \frac{\hat{p} - \mu_{\hat{p}}}{\sigma_{\hat{p}}} = \frac{0.10 - 0.08}{0.01213} \approx \frac{0.02}{0.01213} \approx 1.649<br />\]<br /><br />Next, use the standard normal distribution to find the probability:<br />- Look up the z-score of 1.649 in the standard normal distribution table or use a calculator to find \(P(Z \leq 1.649)\).<br /><br />The cumulative probability \(P(Z \leq 1.649)\) is approximately 0.9505.<br /><br />Therefore, the probability that at least \(10\%\) of the households in the sample own a motorcycle is:<br />\[<br />P(\hat{p} \geq 0.10) = 1 - P(Z \leq 1.649) = 1 - 0.9505 = 0.0495<br />\]<br /><br />So, the probability is approximately **0.0495**.