Harley-Davidson motorcycles make up 14% of all the motorcycles registered in the United States. You plan to interview an SRS of 600 motorcycle owners.

a.) What is the sampling distribution of the proportion of your sample who own Harley's?

b.) How likely is your sample to contain 18.2% or more who own Harley's? How likely is it to contain at least 11.2% Harley owners? Use the 68-95-99.7 rule and your answer to (a).

Thanks for whoever helps~

84 and18.2 percnt not likely but 11.2 is likely

To answer these questions, we need to understand the concept of sampling distribution and the 68-95-99.7 rule.

a.) The sampling distribution of the proportion of your sample who own Harley-Davidson motorcycles can be approximated using the normal distribution. This assumes that your sample is sufficiently large (n ≥ 30) and that the sampling is done randomly and independently.

b.) To determine the likelihood of your sample containing a certain proportion of Harley owners, we can use the 68-95-99.7 rule, also known as the empirical rule or the three-sigma rule. According to this rule:

- Approximately 68% of the data falls within one standard deviation of the mean.
- Approximately 95% of the data falls within two standard deviations of the mean.
- Approximately 99.7% of the data falls within three standard deviations of the mean.

Let's calculate the mean and standard deviation for the proportion of Harley owners in your sample:

Mean = Proportion in the population = 14% = 0.14
Standard Deviation (σ) = sqrt[(p(1-p))/n]
Here, p is the population proportion (0.14) and n is the sample size (600).
Standard Deviation (σ) = sqrt[(0.14 * 0.86) / 600]

Now, let's calculate the Z-scores for the given proportions:

Z-score for 18.2% or more:
Z-score = (X - mean) / σ
Z-score = (0.182 - 0.14) / sqrt[(0.14 * 0.86) / 600]

Z-score for at least 11.2%:
Z-score = (X - mean) / σ
Z-score = (0.112 - 0.14) / sqrt[(0.14 * 0.86) / 600]

Once we have the Z-scores, we can use a Z-table or a statistical software to find the probabilities associated with those Z-scores. The probability will give us the likelihood of observing the given proportions or more extreme values.

Keep in mind that the 68-95-99.7 rule provides a rough approximation, and the actual probabilities may vary slightly depending on the shape of the distribution.