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When we talk about sample proportion, denoted as \( \hat{p} \), we are referring to the fraction of a sample that has a particular trait of interest. In the context of our candy machine problem, \( \hat{p} \) represents the proportion of orange candies in a sample of 25 candies taken from the machine.
The sample proportion is a random variable, meaning it can vary from sample to sample. However, on average, \( \hat{p} \) tends to equal the population proportion \( p \), which is 0.15 in this case. This is because the sample is randomly selected and should, over many samples, reflect the true proportion of the population.
The concept of sample proportion helps us make inferences about the entire population based on just a sample. It's crucial in statistics as it forms the basis for predicting trends and patterns in larger sets.

The standard deviation of the sample proportion's sampling distribution quantifies how much \( \hat{p} \) is expected to fluctuate from the mean in repeated sampling. To calculate it, we use the formula: \[ \sigma_{\hat{p}} = \sqrt{\frac{p(1-p)}{n}} \] where \( p \) is the population proportion, and \( n \) is the sample size.
In our example, substituting \( p = 0.15 \) and \( n = 25 \), we obtain a standard deviation of approximately 0.0735. This reflects the expected variability of the proportion of orange candies when drawing different samples of 25 candies.
Before applying this formula, it’s important to ensure the 10% condition is met. This condition demands that our sample size is less than 10% of the total population, ensuring the samples are sufficiently independent. Given a large candy stock, the condition holds here.

Normal approximation allows us to use the Normal distribution to model our sample proportion distribution, provided certain conditions are met. It's a handy approach because the Normal distribution is well understood and has useful properties.
To use Normal approximation, we check the **Large Counts condition**, requiring two criteria to hold:

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

These criteria ensure that the sample size is large enough to give a reliable approximate distribution.
In the case of our 25 candy sample, these conditions aren't fully satisfied as \( np = 3.75 \), which is less than 10. Thus, the sampling distribution isn't approximately Normal, making Normal approximation inappropriate for such a small sample.

The large sample condition emphasizes the relationship between sample size and the reliability of statistical inferences. Larger samples provide more information and tend to mirror the population more accurately.
For our candy machine example, increasing the sample size from 25 to 225 changes the sampling distribution considerably. The standard deviation of \( \hat{p} \) decreases, leading to more precise results. Specifically, with \( n = 225 \), the standard deviation becomes approximately 0.0266.
Moreover, the larger sample now meets the conditions for Normal approximation, as both \( np = 33.75 \) and \( n(1-p) = 191.25 \) exceed 10. This allows us to comfortably use the Normal model for probability predictions, showcasing the power of having a large enough sample.
Thus, when sample sizes grow, they enhance the reliability and applicability of statistical methods, making large sample conditions pivotal in statistical analyses.