Math

Question Determine the mean and standard deviation of the sampling distribution of the proportion of voters supporting a candidate in a county, given the population proportion and sample size.
Mean: μpundefined=0.2700\mu_{\widehat{p}}=0.2700 Standard error: σpundefined=0.0226\sigma_{\widehat{p}}=0.0226

Studdy Solution

STEP 1

1. The sample proportion pundefined\widehat{p} is a random variable representing the proportion of voters in a sample who support the candidate.
2. The population proportion pp is known, and it is 27%27\% or 0.270.27.
3. The sample size nn is 397397.
4. The sampling distribution of pundefined\widehat{p} can be approximated by a normal distribution since nn is large.
5. The mean of the sampling distribution of pundefined\widehat{p} is equal to the population proportion pp.
6. The standard deviation (standard error) of the sampling distribution of pundefined\widehat{p} can be calculated using the formula σpundefined=p(1p)n\sigma_{\widehat{p}} = \sqrt{\frac{p(1-p)}{n}}.

STEP 2

1. Calculate the mean of the sampling distribution of pundefined\widehat{p}.
2. Calculate the standard error of the sampling distribution of pundefined\widehat{p}.

STEP 3

Determine the mean of the sampling distribution of pundefined\widehat{p}, which is equal to the population proportion pp.
μpundefined=p \mu_{\widehat{p}} = p

STEP 4

Use the formula for the standard error of the sampling distribution of pundefined\widehat{p} to calculate its value.
σpundefined=p(1p)n \sigma_{\widehat{p}} = \sqrt{\frac{p(1-p)}{n}}

STEP 5

Substitute the known values of pp and nn into the formula.
σpundefined=0.27(10.27)397 \sigma_{\widehat{p}} = \sqrt{\frac{0.27(1-0.27)}{397}}

STEP 6

Calculate the value inside the square root.
σpundefined=0.27(0.73)397 \sigma_{\widehat{p}} = \sqrt{\frac{0.27(0.73)}{397}}

STEP 7

Perform the multiplication and division to find the value under the square root.
σpundefined=0.1971397 \sigma_{\widehat{p}} = \sqrt{\frac{0.1971}{397}}

STEP 8

Calculate the value under the square root.
σpundefined=0.0004962 \sigma_{\widehat{p}} = \sqrt{0.0004962}

STEP 9

Take the square root to find the standard error.
σpundefined=0.00049620.0223 \sigma_{\widehat{p}} = \sqrt{0.0004962} \approx 0.0223
The mean of the sampling distribution of pundefined\widehat{p} is 0.270.27 and the standard error is approximately 0.02230.0223 when rounded to four decimal places.
μpundefined=0.27 \mu_{\widehat{p}}=0.27
σpundefined=0.0223 \sigma_{\widehat{p}}=0.0223

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