91Ó°ÊÓ

Understanding the concept of population proportion is essential in statistics. Population proportion, denoted as 'p', represents the fraction or percentage of the total population with a particular characteristic. For instance, in the original exercise, the population proportion is given as 0.84. This means that 84% of the population of 25,000 individuals exhibits the characteristic in question.
Considering population proportion helps in making inferences about a population. For example, if we are studying how many people prefer a certain product, and 84% of our sample prefers it, we can use this proportion to make educated guesses about the entire population.

Sample size, indicated as 'n', refers to the number of observations or data points collected from the population for analysis. In the given problem, the sample size is 1010. Choosing an appropriate sample size is crucial for accuracy as it affects the reliability of the estimates about the population.
A larger sample size generally leads to more precise estimates. This is because it reduces the margin of error, making the data more representative of the population. However, it’s essential to balance between sample size and resources like time and cost. Properly selecting a sample size can make statistical analysis efficient and reliable.

Standard deviation is a measure that quantifies the amount of variation or dispersion in a set of data values. In the context of the sampling distribution of the sample proportion (denoted as \( \hat{p} \)), the standard deviation provides insight into how much the sample proportion is expected to vary from the true population proportion.
The formula for the standard deviation of \( \hat{p} \) is: \[ \sigma_{\hat{p}} = \sqrt{\frac{ p(1 - p)}{n} } \].
Using the values from the exercise: \[ \sigma_{\hat{p}} = \sqrt{\frac{ 0.84(1 - 0.84)}{1010} } = \sqrt{\frac{ 0.84 \cdot 0.16 }{1010} } = \sqrt{ \frac{ 0.1344 }{1010} } = \sqrt{0.000133} \approx 0.0115 \].
This calculation tells us that the standard deviation is approximately 0.0115, indicating how much the sample proportions vary from the actual population proportion.

A normal distribution is a bell-shaped curve that is symmetrical around the mean. This distribution is especially significant in statistics because it represents the distribution of many natural phenomena and measurement errors. When sample sizes are large enough, the sampling distribution of the sample proportion \( \hat{p} \) tends to be normally distributed due to the Central Limit Theorem.
In our exercise, both \( np \) and \( n(1-p) \) are greater than 5, allowing the sampling distribution of \( \hat{p} \) to be approximated by a normal distribution. This approximation lets us use properties of the normal distribution to make inferences about the population proportion. Knowing this is useful for constructing confidence intervals and performing hypothesis tests effectively.
To summarize the sampling distribution of \( \hat{p} \): it is approximately normal with a mean \( \mu_{\hat{p}} = 0.84 \) and standard deviation \( \sigma_{\hat{p}} \approx 0.0115 \).