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The t-distribution is a type of probability distribution that is symmetrical and bell-shaped, similar to the normal distribution but with heavier tails. This characteristic becomes relevant when working with smaller sample sizes because it accounts for the increased variability typically observed in such scenarios.
The t-distribution is especially useful in hypothesis testing when the sample size is small (typically less than 30) and the population standard deviation is unknown. It allows us to estimate the range within which the sample mean would likely fall if the null hypothesis is true.

• The degrees of freedom (df) for the t-distribution depend on the sample size: it is calculated as the number of observations minus one (n-1).
• As the sample size increases, the t-distribution approaches the normal distribution.
• In our exercise, with 10 subjects, we have 9 degrees of freedom (10 - 1 = 9).

Understanding the t-distribution helps in making informed decisions about your hypothesis test, especially when it comes to identifying critical values.

The sample mean, denoted as \( \bar{x} \), is a measure that represents the central tendency of the collected sample data. It provides an estimate of the population mean and is used extensively in hypothesis testing to test claims about the population mean.
To compute the sample mean, you calculate the sum of all sample observations and divide by the number of observations. The formula for the sample mean is: \[ \bar{x} = \frac{\sum X}{n} \]

• In our exercise example, the sum of the sample observations is 1038, and there are 10 subjects. So, \( \bar{x} = \frac{1038}{10} = 103.8 \).
• The sample mean of 103.8 is our best estimate for the true mean of the population in this context.

Using the sample mean in hypothesis testing helps determine if the observed data reflect the population mean hypothesized.

Sample variance is an estimate of the variance of the population from which the sample was drawn. It provides information about the spread or variability of the sample data. It measures how much the data in the sample are spread out around the mean.
The formula for calculating sample variance \( s^2 \) is as follows: \[ s^2 = \frac{\sum X^2 - n(\bar{x})^2}{n-1} \]

• "\( \sum X^2 \)" represents the sum of the squares of each observation in the sample.
• "n" is the number of observations in the sample.
• "\( n-1 \)" is known as degrees of freedom and is used here to eliminate bias in the estimation of population variance.
• In our example, the sample variance was calculated as \( 154.4 \).

This measure is crucial as it helps in understanding the dispersion effects on the sample mean, which is especially important for reliable hypothesis testing.

A two-tailed test is a type of hypothesis test used when we are interested in determining whether there is either a significant increase or decrease in a population parameter compared to a hypothesized value. Unlike one-tailed tests, which assess only one direction, two-tailed tests consider both directions, making them more conservative.
For our scenario, the null hypothesis is that the population mean is equal to 100 \( (H_0: \mu = 100) \), and the alternative hypothesis is that the population mean is not equal to 100 \( (H_1: \mu eq 100) \).

• When using a two-tailed test, the level of significance \( \alpha \) is split between the two tails of the distribution. Typically, 0.05 is divided into 0.025 for each tail.
• This kind of test seeks to find evidence that the sample mean is either significantly higher or lower than the population mean.
• In our example, we found critical values using the t-distribution with 9 degrees of freedom: \( \pm 2.262 \).

Understanding when and how to use a two-tailed test is vital, as it affects how we interpret our test results and determine if our hypothesis stands.