Solved on Dec 12, 2023

Determine the values of $\hat{p}, \hat{q}, n, E$ , and $p$ in a poll of 500 adults about favorite pie, where 11% chose chocolate pie with a margin of error of ±4 percentage points and a 95% confidence level.

STEP 1

Assumptions
1. $\hat{p}$ represents the sample proportion of adults who chose chocolate pie.
2. $\hat{q}$ represents the sample proportion of adults who chose a different pie.
3. $n$ represents the sample size.
4. $E$ represents the margin of error.
5. $p$ represents the population proportion of adults who prefer chocolate pie.
6. The confidence level is given as $95\%$ .
7. $\alpha$ represents the significance level, which is the probability that the true proportion lies outside the confidence interval.

STEP 2

Identify the value of $\hat{p}$ , which is the sample proportion of adults who chose chocolate pie.
$\hat{p} = 11\%$

STEP 3

Convert the percentage to a decimal value for $\hat{p}$ .
$\hat{p} = \frac{11}{100} = 0.11$

STEP 4

Calculate the value of $\hat{q}$ , which is the sample proportion of adults who chose a different pie. Since $\hat{p} + \hat{q} = 1$ , we have:
$\hat{q} = 1 - \hat{p}$

STEP 5

Plug in the value of $\hat{p}$ to find $\hat{q}$ .
$\hat{q} = 1 - 0.11 = 0.89$

STEP 6

Identify the value of $n$ , which is the sample size.
$n = 500$

STEP 7

Identify the value of $E$ , which is the margin of error.
$E = \pm 4\%$

STEP 8

Convert the percentage to a decimal value for $E$ .
$E = \frac{4}{100} = 0.04$

STEP 9

Recognize that $p$ is the true population proportion, which we are estimating with our sample proportion $\hat{p}$ . Since we do not have the true population proportion, we use $\hat{p}$ as our best estimate.
$p \approx \hat{p} = 0.11$

STEP 10

Determine the value of $\alpha$ given the confidence level of $95\%$ . The confidence level indicates that $95\%$ of the time, the true proportion $p$ will lie within the confidence interval. The significance level $\alpha$ is the probability that the true proportion lies outside the confidence interval.
$\alpha = 1 - \text{confidence level}$

STEP 11

Convert the confidence level to a decimal and calculate $\alpha$ .
$\alpha = 1 - 0.95 = 0.05$
The value of $\hat{p}$ is $0.11$ . The value of $\hat{q}$ is $0.89$ . The value of $n$ is $500$ . The value of $E$ is $\pm 0.04$ . The value of $p$ is approximately $0.11$ . The value of $\alpha$ is $0.05$ .

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Determine the values of $\hat{p}, \hat{q}, n, E$ , and $p$ in a poll of 500 adults about favorite pie, where 11% chose chocolate pie with a margin of error of ±4 percentage points and a 95% confidence level.

STEP 1

STEP 2

Identify the value of $\hat{p}$ , which is the sample proportion of adults who chose chocolate pie.
$\hat{p} = 11\%$

STEP 3

Convert the percentage to a decimal value for $\hat{p}$ .
$\hat{p} = \frac{11}{100} = 0.11$

STEP 4

Calculate the value of $\hat{q}$ , which is the sample proportion of adults who chose a different pie. Since $\hat{p} + \hat{q} = 1$ , we have:
$\hat{q} = 1 - \hat{p}$

STEP 5

Plug in the value of $\hat{p}$ to find $\hat{q}$ .
$\hat{q} = 1 - 0.11 = 0.89$

STEP 6

Identify the value of $n$ , which is the sample size.
$n = 500$

STEP 7

Identify the value of $E$ , which is the margin of error.
$E = \pm 4\%$

STEP 8

Convert the percentage to a decimal value for $E$ .
$E = \frac{4}{100} = 0.04$

STEP 9

Recognize that $p$ is the true population proportion, which we are estimating with our sample proportion $\hat{p}$ . Since we do not have the true population proportion, we use $\hat{p}$ as our best estimate.
$p \approx \hat{p} = 0.11$

STEP 10

STEP 11

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Determine the values of p^,q^,n,E\hat{p}, \hat{q}, n, Ep^​,q^​,n,E, and ppp in a poll of 500 adults about favorite pie, where 11% chose chocolate pie with a margin of error of ±4 percentage points and a 95% confidence level.

STEP 1

STEP 2

Identify the value of p^\hat{p}p^​, which is the sample proportion of adults who chose chocolate pie.p^=11%\hat{p} = 11\%p^​=11%

STEP 3

Convert the percentage to a decimal value for p^\hat{p}p^​.p^=11100=0.11\hat{p} = \frac{11}{100} = 0.11p^​=10011​=0.11

STEP 4

Calculate the value of q^\hat{q}q^​, which is the sample proportion of adults who chose a different pie. Since p^+q^=1\hat{p} + \hat{q} = 1p^​+q^​=1, we have:q^=1−p^\hat{q} = 1 - \hat{p}q^​=1−p^​

STEP 5

Plug in the value of p^\hat{p}p^​ to find q^\hat{q}q^​.q^=1−0.11=0.89\hat{q} = 1 - 0.11 = 0.89q^​=1−0.11=0.89

STEP 6

Identify the value of nnn, which is the sample size.n=500n = 500n=500

STEP 7

Identify the value of EEE, which is the margin of error.E=±4%E = \pm 4\%E=±4%

STEP 8

Convert the percentage to a decimal value for EEE.E=4100=0.04E = \frac{4}{100} = 0.04E=1004​=0.04

STEP 9

Recognize that ppp is the true population proportion, which we are estimating with our sample proportion p^\hat{p}p^​. Since we do not have the true population proportion, we use p^\hat{p}p^​ as our best estimate.p≈p^=0.11p \approx \hat{p} = 0.11p≈p^​=0.11

STEP 10

STEP 11

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Determine the values of $\hat{p}, \hat{q}, n, E$ , and $p$ in a poll of 500 adults about favorite pie, where 11% chose chocolate pie with a margin of error of ±4 percentage points and a 95% confidence level.

Identify the value of $\hat{p}$ , which is the sample proportion of adults who chose chocolate pie.
$\hat{p} = 11\%$

Convert the percentage to a decimal value for $\hat{p}$ .
$\hat{p} = \frac{11}{100} = 0.11$

Calculate the value of $\hat{q}$ , which is the sample proportion of adults who chose a different pie. Since $\hat{p} + \hat{q} = 1$ , we have:
$\hat{q} = 1 - \hat{p}$

Plug in the value of $\hat{p}$ to find $\hat{q}$ .
$\hat{q} = 1 - 0.11 = 0.89$

Identify the value of $n$ , which is the sample size.
$n = 500$

Identify the value of $E$ , which is the margin of error.
$E = \pm 4\%$

Convert the percentage to a decimal value for $E$ .
$E = \frac{4}{100} = 0.04$

Recognize that $p$ is the true population proportion, which we are estimating with our sample proportion $\hat{p}$ . Since we do not have the true population proportion, we use $\hat{p}$ as our best estimate.
$p \approx \hat{p} = 0.11$