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The sample proportion, often denoted as \( \hat{p} \), is the ratio of elements within a sample that have a specific characteristic. In this case, it's the proportion of orange candies in the sample drawn from the candy machine. The sample proportion serves as an unbiased estimator of the population proportion, meaning on average, it correctly reflects the proportion in the entire population.
The formula for determining \( \hat{p} \) is:

• \( \hat{p} = \frac{x}{n} \)

- where \( x \) is the count of orange candies in the sample, and \( n \) is the sample size.
This value provides insight into the underlying proportion of the entire population of candies when taking random samples. However, variability exists since each sample may not perfectly mirror the population.

The standard deviation of the sampling distribution of \( \hat{p} \) measures the variability of the sample proportion around the population proportion \( p \). It tells us how much we can expect \( \hat{p} \) to vary from sample to sample.
To calculate this, use the formula:

• \[ \sigma_{\hat{p}} = \sqrt{\frac{p(1-p)}{n}} \]

In our example, with \( p = 0.45 \) and \( n = 25 \), the standard deviation is approximately \( 0.0995 \). This value indicates that the sample proportion will typically vary by about 0.0995 from the true proportion in the population.
Lower standard deviation implies that repeated sampling would yield sample proportions closer to the true population proportion.

The normal condition is crucial to determine if the sampling distribution of \( \hat{p} \) can be approximated by a normal distribution. This is important because a normal distribution makes it easier to apply statistical inference methods.
To check if the normal condition is satisfied, we ensure:

• \( np > 10 \)
• \( n(1-p) > 10 \)

In our case, both calculations:\( np = 25 \times 0.45 = 11.25 \) and \( n(1-p) = 25 \times 0.55 = 13.75 \), are greater than 10, thus confirming that the sampling distribution is approximately normal. This supports analyses that require normality assumptions, such as hypothesis tests and confidence intervals.

The 10% condition ensures that the sample size is sufficiently small relative to the population to allow the model assumptions to hold. This condition helps maintain the independence of trials within the sample.
Specifically, the sample size, \( n \), should be less than 10% of the population size, or \( n < 0.10N \) where \( N \) is the population size.
In this context, if the candy machine contains significantly more than 250 candies, then our sample of 25 satisfies this condition. This means any outcomes from our sample should reliably reflect those possible in the entire population without being affected by the depletion of available options.