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The mean of a sampling distribution is a crucial concept to understand when analyzing proportion data from surveys or experiments. For the exercise at hand, we are specifically looking at adults in Great Britain who report running water in the shower for 6 to 10 minutes, with 30% doing so according to the survey data.

This mean of the sampling distribution, represented as \( \mu_{\hat{p}} \), represents the average proportion of people expected to meet this criterion in any random sample. Conveniently, in the context of a sampling distribution based on a known population proportion, \( \mu_{\hat{p}} \) is equal to the population proportion, which is 0.30 in this example.

• This means if we took many random samples, on average 30% of each sample would also indicate they run the water for 6 to 10 minutes.
• Therefore, the mean acts as a balancing point for all potential sample outcomes based on the given population behavior.

This is foundational when it comes to predicting the behavior of larger groups based on small samples.

The standard deviation of a sampling distribution, symbolized as \( \sigma_{\hat{p}} \), is key to understanding the variability in sample proportions. It tells us how much the proportions in individual samples vary from the overall population proportion.

To determine \( \sigma_{\hat{p}} \), we use the formula for a standard deviation of a sample proportion:\[\sigma_{\hat{p}} = \sqrt{ \frac{p(1-p)}{n} }\]Here, the values we substitute are:

• \( p = 0.30 \)
• \( n = 180 \)

This simplifies into:\[\sigma_{\hat{p}} = \sqrt{ \frac{0.30 \times 0.70}{180} }\]

• By performing the calculation, we can see how much each sample varies around the mean proportion.
• Understanding this variance helps in predicting the likelihood of sample proportions differing vastly from the population proportion.

This knowledge assists in determining how tight or spread out the sample outcomes might be.

To explore the shape of the sampling distribution, particularly if it is close to a normal distribution, we need to evaluate conditions that allow us to make this approximation.

The rule of thumb for normal approximation is applied when both \( np > 5 \) and \( n(1-p) > 5 \) are satisfied:

• \( np = 54 \)
• \( n(1-p) = 126 \)

Since both these values are significantly greater than 5, it implies that our sampling distribution \( \hat{p} \) can be well-approximated by a normal distribution.

• This facilitates easier predictions and statistical inference, as many statistical tools and methods rely on normal distribution assumptions.
• The central limit theorem supports this approximation, ensuring that even if the population distribution isn’t normal, the sampling distribution of the sample mean will tend toward a normal distribution as the sample size large enough.

Understanding this gives you the power to apply normal distribution models when working with sample proportions.