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The Student's t-distribution is a probability distribution that is particularly useful when dealing with small sample sizes. It is similar to the normal distribution in shape but has heavier tails. This means it has a higher probability of values further away from the mean, which accounts for the higher variability expected with smaller samples.

You might wonder why it's called the Student's t-distribution. Well, it got its name from a statistician named William Sealy Gosset who published under the pseudonym "Student". The t-distribution is vital because it allows us to make inferences about a population mean when the population standard deviation is unknown and the sample size is small.

In our case, with a sample size of 16, using the Student's t-distribution helps accurately estimate the confidence interval, ensuring our results are reliable despite not having a large sample.

Sample size plays a critical role in statistical confidence and reliability. When the sample size is small, typically less than 30, we often rely on the Student's t-distribution for estimating the population mean. This is because the t-distribution adapts to the smaller sample size's higher variability.

In this exercise, the sample size is 16, which is considered small. For a small sample, the central limit theorem (which states that the sampling distribution of the sample mean will be normal) does not adequately apply unless the sample comes from a normal distribution. Hence, the use of the t-distribution is appropriate to capture the true variability.

If our sample were larger, say over 30, we might opt to use the standard normal distribution (z-distribution) as it gives similarly reliable estimates in larger sample sizes.

A symmetric distribution is one where the left and right sides of the distribution are mirror images of each other. This characteristic is crucial when using the Student's t-distribution because it does not assume normality but does assume a symmetric bell shape similar to the normal distribution.

In the given problem, it's stated that the distribution is mound-shaped and symmetric. This description ensures that the t-distribution can be applied without issue. When the distribution is skewed or not symmetric, the estimates from a t-distribution might become biased.

Symmetry helps in keeping the assumptions of the statistical methods intact, lending more reliability to the confidence interval derived from such data.

Selecting the correct t-score is a vital step in calculating the confidence interval. The t-score corresponds to the desired confidence level and the degrees of freedom, which is one less than the sample size (i.e., -1).

For a 90% confidence interval and 15 degrees of freedom (since our sample size is 16), we refer to the t-distribution table to find the required t-score. In our example, this value is approximately 1.753. This critical value accounts for the tails of the distribution being heavier, catering to the variability in smaller samples.

Choosing the right t-score ensures that our confidence interval appropriately covers the desired proportion of the data, representing a 90% likelihood that the true population mean is within this interval.

Calculating the confidence interval involves combining all the elements we've discussed so far. The formula used is \[ \text{CI} = \bar{x} \pm t_{\alpha/2} \times \frac{s}{\sqrt{n}} \], where \( \bar{x} \) is the sample mean, \( t_{\alpha/2} \) is the t-score, \( s \) is the sample standard deviation, and \( n \) is the sample size.

In our example:

• The sample mean \( \bar{x} \) is 10.
• The sample standard deviation \( s \) is 2.
• The t-score (for 90% confidence and 15 degrees of freedom) is 1.753.
• The confidence interval then becomes \( 10 \pm 1.753 \times \frac{2}{\sqrt{16}} = 10 \pm 0.8765 \).

Thus, the confidence interval is approximately \([9.1235, 10.8765]\).

This range indicates that we have a 90% confidence that the true population mean \( \mu \) lies within \( 9.1235 \) to \( 10.8765 \). This means, if we repeated the sampling process many times, about 90% of the intervals we compute would contain the true population mean.