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When constructing confidence intervals, especially with small sample sizes, a t-distribution is often used rather than the normal distribution. This is because the t-distribution accounts for additional variability due to the smaller sample size.
T-distributions are a family of distributions that are similar in shape to the normal distribution but have thicker tails. This property allows them to be more forgiving with small sample sizes, capturing the increased uncertainty in estimating population parameters.
Key details about t-distributions are:

• As the sample size increases, the t-distribution approaches the normal distribution.
• The degrees of freedom ( \(df\) ) for a t-distribution is usually calculated as ( \(n-1\) ) where ( \(n\) ) is the sample size.
• For a larger \(df\), the t-distribution becomes closer to the standard normal distribution (Z-distribution).

In the given exercise, since the sample size ( \(n = 31\) ) is not very large, a t-distribution with 30 degrees of freedom was used to find the critical t-value, which was approximately 2.042 for a 95% confidence interval.

The standard error (SE) measures the variability of the sample mean from the actual population mean. It provides an estimate of the expected difference between the sample mean we calculate and the true population mean.
The formula for calculating the standard error of the mean is: \[SE = \frac{s}{\sqrt{n}}\] where ( \(s\) ) is the sample standard deviation and ( \(n\) ) is the sample size.
A smaller standard error indicates that our sample mean is a more accurate estimate of the population mean. In our example, the calculated standard error ( \(SE = 3,030.12\) ) reveals the average expected error in predicting the population mean using our sample data.
Key points about standard error:

• Larger samples tend to have smaller standard errors, implying more precise estimates.
• Standard error decreases as the sample size increases because it becomes a better representation of the population.
• It plays a crucial role in constructing confidence intervals, helping determine how wide or narrow the interval will be.

Constructing a confidence interval using a t-distribution assumes that the population from which the sample is drawn is normally distributed. When the sample size is small, this assumption becomes even more critical because the t-distribution relies on this normality to provide accurate confidence intervals.
However, when the population distribution is not known or is not normal, the reliability of the confidence interval may be questioned, especially with a smaller sample size like 31. Yet, thankfully, the Central Limit Theorem offers some reassurance:

• With larger sample sizes (typically over 30), the sample means tend to form a normal distribution, even if the underlying population distribution is not normal.
• This principle allows us to use normality-based methods like the t-distribution to estimate confidence intervals effectively.

If the normality assumption is violated in a smaller sample, then the confidence interval might not accurately capture the true population mean. However, with larger sample sizes, this concern becomes less significant, offering more reliability in the interval estimate.