91Ó°ÊÓ

The sample mean is a fundamental concept in statistics. It provides a single value summary of a data set, offering a measure of the central tendency. In the context of our exercise, the sample mean was calculated from the responses of 10 students regarding how often they feel sad.
To find the sample mean (\(\bar{x}\)), you sum up all the data points and divide by the number of points. For our data, it was computed as follows:

• Sum of responses: 0 + 0 + 1 + 0 + 7 + 2 + 1 + 0 + 0 + 3 = 14
• Number of responses: 10
• Sample mean: \(\bar{x} = \frac{14}{10} = 1.4\)

This sample mean of 1.4 provides an average estimate of how often these students feel sad within the context of our study.

Standard deviation is a measure of the dispersion or spread in a set of data. It tells us how much the data values deviate from the mean on average. In simple terms, it shows how spread out the numbers are in a dataset.
Calculating the sample standard deviation involves several steps:

• Find the deviation of each data point from the sample mean.
• Square these deviations to remove negative values.
• Sum the squared deviations and divide by \(n-1\), where \(n\) is the number of data points.
• Take the square root of the result.

In the given exercise, the sample standard deviation was approximately 1.673. This value indicates variability in students' responses about feeling sad compared to the mean.

The t-distribution is an essential concept used when estimating population parameters with a smaller sample size (\(n < 30\)). It accounts for the increased variability inherent in small samples. The shape is similar to a normal distribution but has heavier tails.
For constructing a confidence interval, we use a t-value derived from the t-distribution. This value depends on the desired level of confidence and the degrees of freedom (\(df\), calculated as \(n - 1\)). In this exercise, with \(df = 9\) for 10 data points and a 90% confidence level, the corresponding t-value is approximately 1.833.
This t-value is crucial in determining how much uncertainty there is in our sample mean estimation.

The margin of error quantifies the uncertainty in our estimate of the population parameter, based on the sample. It's the range above and below the sample mean that the true population mean is expected to fall into with a specified confidence level.
In the context of our exercise, the margin of error (\(ME\)) is calculated using the formula: \(ME = t \cdot \frac{s}{\sqrt{n}}\).

• \(t\) is the t-value from the t-distribution table.
• \(s\) is the sample standard deviation.
• \(n\) is the number of observations.

With our numbers, the margin of error was about 0.970. Adding and subtracting this from the sample mean of 1.4 gives the confidence interval. This implies we are quite sure the true population mean lies in this interval, accounting for the uncertainty.