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When comparing two different populations, like males and females, one can calculate the difference in their population means to understand how they differ in a specific characteristic, such as the number of close friends. In this case, we are comparing the means of close friends reported by females and males. The estimated difference between these means is calculated using the formula:
\[ \hat{\mu}_1 - \hat{\mu}_2 \]
where \( \hat{\mu}_1 \) is the mean number of friends for males (8.9) and \( \hat{\mu}_2 \) is the mean number for females (8.3). Substituting these values, we get:
- Estimated mean difference = \( 8.9 - 8.3 = 0.6 \)
This result suggests that, on average, males have about 0.6 more close friends than females. However, this is just an estimate and not a definitive proof. The confidence interval helps us understand the precision of this estimate.

The normality assumption refers to the belief that the data in question follows a normal distribution, a crucial requirement in many statistical methods, especially when calculating confidence intervals. Normal distribution is symmetric and follows a bell curve shape. It assumes most data points cluster around the mean, with fewer points at the extremes.
For the dataset in the problem, we calculated standard deviations of 15.6 for females and 15.5 for males, which are quite large compared to their respective means (8.3 and 8.9). This large spread hints that the data might not be normally distributed since normal distribution typically expects smaller standard deviations relative to the mean.
When the data doesn't adhere to normality, confidence interval estimates may become unreliable. However, if the sample size is large, the central limit theorem can often offset deviations from normality.

The Central Limit Theorem (CLT) is a cornerstone of probability theory and statistics. It implies that, regardless of the distribution of a population, the sampling distribution of the sample mean will be approximately normally distributed if the sample size is large enough.
Given that the surveyed group includes 486 females and 354 males, these are sizeable samples, which suggests the CLT is applicable. Therefore, even if the data does not follow a normal distribution due to large standard deviations, we can still approximate the distribution of the sample means as normal.
This property of the CLT is extremely useful when performing statistical tests or constructing confidence intervals, especially when assessing differences in population means. In this context, even with non-normal data, the large sample sizes allow us to still rely on the confidence interval estimates to inform about the population mean differences.