Questions: A marketing team is targeting people who might buy a hybrid car. In their city, with a population of 30,000 people, 3,170 people either drive a hybrid car or have indicated on a recent survey that they would be interested in driving one. Find the population proportion, as well as the mean and standard deviation of the sampling distribution for samples of size n=121. Round all answers to 3 decimal places. Provide your answer below: p= μp= σp=

A marketing team is targeting people who might buy a hybrid car. In their city, with a population of 30,000 people, 3,170 people either drive a hybrid car or have indicated on a recent survey that they would be interested in driving one. Find the population proportion, as well as the mean and standard deviation of the sampling distribution for samples of size n=121. Round all answers to 3 decimal places.

Provide your answer below:
p=
μp=
σp=

Transcript text: A marketing team is targeting people who might buy a hybrid car. In their city, with a population of 30,000 people, 3,170 people either drive a hybrid car or have indicated on a recent survey that they would be interested in driving one. Find the population proportion, as well as the mean and standard deviation of the sampling distribution for samples of size $n=121$. Round all answers to 3 decimal places. Provide your answer below: \[ \begin{array}{l} p=\square \\ \mu_{\hat{p}}=\square \\ \sigma_{\hat{p}}=\square \end{array} \]

Solution

Solution Steps

Step 1: Calculate the Population Proportion

The population proportion $ p $ is calculated as follows:

\[ p = \frac{\text{Number of people interested in hybrid cars}}{\text{Total population}} = \frac{3170}{30000} \approx 0.106 \]

Step 2: Mean of the Sampling Distribution

The mean of the sampling distribution of the sample proportion $ \mu_{\hat{p}} $ is equal to the population proportion:

\[ \mu_{\hat{p}} = p \approx 0.106 \]

Step 3: Standard Deviation of the Sampling Distribution

The standard deviation of the sampling distribution of the sample proportion $ \sigma_{\hat{p}} $ is calculated using the formula:

\[ \sigma_{\hat{p}} = \sqrt{\frac{p(1 - p)}{n}} = \sqrt{\frac{0.106(1 - 0.106)}{121}} \approx 0.028 \]

Final Answer

\[ \begin{array}{l} p = 0.106 \\ \mu_{\hat{p}} = 0.106 \\ \sigma_{\hat{p}} = 0.028 \end{array} \]

Thus, the final boxed answer is: \[ \boxed{ \begin{array}{l} p = 0.106 \\ \mu_{\hat{p}} = 0.106 \\ \sigma_{\hat{p}} = 0.028 \end{array} } \]

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