91Ó°ÊÓ

When it comes to statistical analyses, the concept of 'degrees of freedom' is a crucial one. It refers to the number of independent values or quantities which can be assigned to a statistical distribution. In most cases, the degrees of freedom in a statistical calculation are equal to the number of values that are free to vary. Imagine having a set of numbers with a fixed mean; if you already know all but one of those numbers, the last one can be determined to maintain the mean. Therefore, it only has one degree of freedom. Similarly, when working with a sample of size n, the degrees of freedom for calculations related to the sample mean is usually n-1, because the total of the sample values is constrained by the sample mean.
To make it relatable, consider a real-life scenario where a teacher has ten apples to distribute among nine students. She can give out apples to eight students in any manner she wishes, but the ninth student's number of apples is no longer a free choice—it’s determined by the remaining apples. This simple example mirrors the rationale behind degrees of freedom in statistics, which explains why, as in our exercise, the sample size of 18 leads to 17 degrees of freedom (\(df = n - 1 = 18 - 1 = 17\)).

The process of making decisions or predictions about a population based on sample data is known as statistical inference. When it comes to the inference for the sample mean, this involves using sample data to estimate the population mean. Since we don't have access to the whole population, we rely on our sample to provide insight. However, we must account for variability in our estimate; this is where the t-distribution comes into play when the sample size is small or the population standard deviation is unknown.
By using the t-distribution, we can determine confidence intervals or conduct hypothesis tests regarding the population mean. In our exercise, the t-distribution is employed to find the endpoints of the confidence interval. This tells us within what range we can expect to find the population mean, based on our sample, with a certain level of confidence. A key factor in such inference is the aforementioned degrees of freedom, which affects the shape of the t-distribution and the accuracy of our estimates.
So, in essence, inference for the sample mean is like piecing together a puzzle with some missing pieces. We use the pieces we do have (our sample) to estimate what the full picture (population mean) is likely to be. In our problem, we're determining how far out the puzzle's edges go, with \(1\%\) uncertainty on each end.

The normal distribution, often called the bell curve due to its shape, is a common probability distribution that's symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean. In fact, the farther away from the mean, the lower the frequency of occurrence. Notably, in a normal distribution, the mean, median, and mode are all equal.
A standard normal distribution is a normal distribution with a mean of zero and a standard deviation of one. It is a foundational concept in statistics and underpins many statistical techniques, including the t-distribution. When the sample size is sufficiently large, usually more than 30, the central limit theorem tells us that the sampling distribution of the sample mean will be approximately normally distributed, even if the underlying population distribution is not.
Returning to our exercise, we assume the sample is from a distribution that is 'reasonably normally distributed’, which means it's similar enough to a normal distribution for purposes of the statistical methods being used. When we have smaller sample sizes or an unknown population standard deviation, the normal distribution isn't a proper fit anymore. This is where the t-distribution steps in, aptly adjusting for the increased uncertainty and providing a more accurate tool for our inference about the sample mean.