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 , the sample proportion of households that own a motorcycle.### (a) Describe the shape, center, and variability of the sampling distribution of .1. **Shape**: The shape of the sampling distribution of is approximately normal if both and \(n(1-p)\) are greater than 10. Given: - - Calculate: Since both and \(n(1-p) = 460\) are greater than 10, the shape of the sampling distribution is approximately **normal**.2. **Center**: The mean of the sampling distribution of is equal to the population proportion . 3. **Variability**: The standard deviation of the sampling distribution of is given by: ### (b) Find the probability that at least of the households in the sample own a motorcycle.We want to find \(P(\hat{p} \geq 0.10)\).First, calculate the z-score for : Next, use the standard normal distribution to find the probability:- Look up the z-score of 1.649 in the standard normal distribution table or use a calculator to find \(P(Z \leq 1.649)\).The cumulative probability \(P(Z \leq 1.649)\) is approximately 0.9505.Therefore, the probability that at least of the households in the sample own a motorcycle is: So, the probability is approximately **0.0495**.