If it is known that the population proportion of car parks in a particular city that offer early discount fees is 25%, which of the following best describes the mean and standard deviation of the sampling distribution of the sample proportion of car parks that offer early discount fees for samples taken of size 30?

Question

BlackTom AI · Accepted Answer

This question asks about the mean and standard deviation of the sampling distribution of the sample proportion, p-hat, when sampling 30 car parks with a true population proportion p = 0.25.

First, recall the key formulas: for a binomial (or proportion) scenario, the sampling distribution of p-hat has mean μ_p̂ = p and standard deviation σ_p̂ = sqrt[ p(1 - p) / n ].

Plugging in the given values: p = 0.25 and n = 30.
- Mean: μ_p̂ = p = 0.25.
- Standard deviation: σ_p̂ = sqrt[ 0.25 × 0.75 / 30 ] = sqrt[ 0.1875 / 30 ] = sqrt[ 0.00625 ] ≈ 0.079.

Therefore, the correct description of the sampling distribution is that its mean is 0.25 and its standard deviation is about 0.079.

If any answer choices mentioned those two numbers, they would be describing the same distribution. However, since the provided list of options is missing here, we can’t compare against other alternatives, but the calculation above shows the standard deviation as approximately 0.079 and the mean as 0.25 are the correct values for n = 30 and p = 0.25.

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