Suppose the true proportion of voters in the county who support a specific candidate is 0.48. Consider the sampling distribution for the proportion of supporters with sample size n = 177. What is the mean of this distribution? What is the standard error (i.e. the standard deviation) of this sampling distribution, rounded to three decimal places?

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**Title: Understanding Sampling Distributions in Election Surveys**

**Scenario:**
Suppose the true proportion of voters in the county who support a specific candidate is 0.48. Consider the sampling distribution for the proportion of supporters with sample size \(n = 177\).

**Questions:**

1. **What is the mean of this distribution?**
   
   (Input Box for Response)

2. **What is the standard error (i.e., the standard deviation) of this sampling distribution, rounded to three decimal places?**

   (Input Box for Response)

**Explanation:**

In this scenario, students are asked to calculate key parameters of a sampling distribution. The true proportion of supporters, denoted by \( p \), is given as 0.48. The sample size \( n \) is 177. 

**Mean of the Sampling Distribution:**
The mean of the sampling distribution of the sample proportion (\( \hat{p} \)) is equal to the true proportion \( p \). Hence, in this case, it would be:

\[ \text{Mean} = p = 0.48 \]

**Standard Error of the Sampling Distribution:**
The standard error (SE) of the sample proportion can be calculated using the formula:

\[ SE = \sqrt{ \frac{p(1-p)}{n} } \]

Substituting the given values:

\[ SE = \sqrt{ \frac{0.48 \times (1 - 0.48)}{177} } \]

Perform the calculation to find the standard error.

Students are to enter their responses in the provided input boxes. The calculations will help in understanding how sample size and proportion affect the variability of survey results and the reliability of estimates in population studies.
Transcribed Image Text:**Title: Understanding Sampling Distributions in Election Surveys** **Scenario:** Suppose the true proportion of voters in the county who support a specific candidate is 0.48. Consider the sampling distribution for the proportion of supporters with sample size \(n = 177\). **Questions:** 1. **What is the mean of this distribution?** (Input Box for Response) 2. **What is the standard error (i.e., the standard deviation) of this sampling distribution, rounded to three decimal places?** (Input Box for Response) **Explanation:** In this scenario, students are asked to calculate key parameters of a sampling distribution. The true proportion of supporters, denoted by \( p \), is given as 0.48. The sample size \( n \) is 177. **Mean of the Sampling Distribution:** The mean of the sampling distribution of the sample proportion (\( \hat{p} \)) is equal to the true proportion \( p \). Hence, in this case, it would be: \[ \text{Mean} = p = 0.48 \] **Standard Error of the Sampling Distribution:** The standard error (SE) of the sample proportion can be calculated using the formula: \[ SE = \sqrt{ \frac{p(1-p)}{n} } \] Substituting the given values: \[ SE = \sqrt{ \frac{0.48 \times (1 - 0.48)}{177} } \] Perform the calculation to find the standard error. Students are to enter their responses in the provided input boxes. The calculations will help in understanding how sample size and proportion affect the variability of survey results and the reliability of estimates in population studies.
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