Questions: Smartphones: A poll agency reports that 64% of teenagers aged 12-17 own smartphones. A random sample of 65 teenagers is drawn. Round your answers to at least four decimal places as needed. Part: 0 / 6 Part 1 of 6 (a) Find the mean μᵖ̂. The mean μₚ is 0.64. Part: 1 / 6 Part 2 of 6 (b) Find the standard deviation σᵖ̂ The standard deviation σᵖ̂ is .

Smartphones: A poll agency reports that 64% of teenagers aged 12-17 own smartphones. A random sample of 65 teenagers is drawn. Round your answers to at least four decimal places as needed. Part: 0 / 6 Part 1 of 6 (a) Find the mean μᵖ̂. The mean μₚ is 0.64. Part: 1 / 6 Part 2 of 6 (b) Find the standard deviation σᵖ̂ The standard deviation σᵖ̂ is .
Transcript text: Smartphones: A poll agency reports that $64 \%$ of teenagers aged $12-17$ own smartphones. A random sample of 65 teenagers is drawn. Round your answers to at least four decimal places as needed. Part: $0 / 6$ $\square$ Part 1 of 6 (a) Find the mean $\mu_{\hat{p}}$. The mean $\mu_{p}$ is 0.64 . $\square$ Part: $1 / 6$ $\square$ Part 2 of 6 (b) Find the standard deviation $\sigma_{\hat{p}}$ The standard deviation $\sigma_{\hat{p}}$ is $\square$. $\square$
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Solution

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Solution Steps

Step 1: Calculate the Mean \( \mu_{\hat{p}} \)

The mean of the sample proportion \( \mu_{\hat{p}} \) is given by the population proportion \( p \). Therefore, we have:

\[ \mu_{\hat{p}} = p = 0.64 \]

Step 2: Calculate the Standard Deviation \( \sigma_{\hat{p}} \)

The standard deviation of the sample proportion \( \sigma_{\hat{p}} \) is calculated using the formula:

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

Substituting the values \( p = 0.64 \) and \( n = 65 \):

\[ \sigma_{\hat{p}} = \sqrt{\frac{0.64 \times (1 - 0.64)}{65}} = \sqrt{\frac{0.64 \times 0.36}{65}} \approx 0.0595 \]

Final Answer

The results are:

  • Mean \( \mu_{\hat{p}} = 0.6400 \)
  • Standard Deviation \( \sigma_{\hat{p}} = 0.0595 \)

Thus, the final answers are:

\[ \boxed{\mu_{\hat{p}} = 0.6400} \] \[ \boxed{\sigma_{\hat{p}} = 0.0595} \]

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