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When we talk about sample proportion, we are referring to the proportion of individuals in a sample who exhibit a certain characteristic. In the context of this exercise, the sample proportion, denoted as \( \hat{p} \), represents those who attended church in the given sample of 1785 adults. This proportion helps us understand the characteristics of the entire population based on the sampled data.
The sample proportion is calculated by dividing the number of people who have the characteristic by the total number of people in the sample. It's important because it serves as an estimate for the population proportion \( p \), especially in statistical inference.

• If everyone in the sample attends church, \( \hat{p} = 1 \).
• If no one in the sample attends, \( \hat{p} = 0 \).
• Values in between describe varying levels of attendance.

Understanding sample proportion is key to making predictions about larger groups, given only as limited information.

Standard deviation is a measure of how spread out numbers are around a mean. In the context of sampling distribution, it tells us how much the sample proportions vary from the true population proportion, \( p \).
For a sample proportion, the standard deviation is found using the formula: \[\sigma_{\hat{p}} = \sqrt{\frac{p(1-p)}{n}} \]Here, \( n \) is the sample size, and \( p \) is the assumed population proportion. This formula tells us how much variability to expect in the sample proportion if we were to take many samples.
This measure is crucial to determine how much faith we should have in our sample proportion as an estimate of the population proportion. Less variability (a smaller standard deviation) means we have a more precise estimate.

A normal distribution is a fundamental statistical concept frequently appearing in probability and statistics. It's often represented as a bell-shaped curve. In a normal distribution, most observations cluster around the mean, and probabilities for values taper off symmetrically as you move away from the mean.
For a sampling distribution to be reasonably approximated as normal, certain conditions—like the Central Limit Theorem—must justify it. Here, we use the Large Counts condition, which says:

• \( np \geq 10\)
• \( n(1-p) \geq 10\)

This means that both the number of successes (attendees) and failures (non-attendees) must be sufficiently large.
Meeting these conditions allows us to assume the sampling distribution of \( \hat{p} \) is normal, enabling the use of z-scores and other tools of normal probability in further analysis.

Calculating probability is about determining how likely an event is to occur. In the exercise, we want to know the probability that the sample proportion of adults who attended church is 44% or more, under the assumption that the true population proportion is 40%.
We begin by calculating a z-score to understand how far \( \hat{p} \) (the observed proportion) is from \( p \) (the claimed proportion). The formula for the z-score is:\[z = \frac{\hat{p} - p}{\sigma_{\hat{p}}}\]Given \( \hat{p} = 0.44 \) and \( p = 0.40 \), inserting these values and the standard deviation calculated earlier tells us how extreme our sample result is. A high z-score corresponds to a low probability of the sample proportion \( \hat{p} \) occurring if the newspaper report is true.
If the probability is very low, this suggests that the sample provides evidence against the claimed population proportion of 40%, implying significant statistical findings in consideration of the poll’s results.