Degrees of Freedom In general, what does “degrees of freedom” refer to? For the sample data described in Exercise 7 “Requirements,” find the number of degrees of freedom, assuming that you want to construct a confidence interval estimate of u using the t distribution.

Verified step by step guidance

1. Step 1: Understand the concept of 'degrees of freedom' (df). In statistics, degrees of freedom refer to the number of independent values or quantities that can vary in an analysis without violating any given constraints. For example, in a sample of size n, the degrees of freedom are often related to the number of values that are free to vary after certain parameters (like the mean) are fixed.
2. Step 2: Identify the context of the problem. Here, we are constructing a confidence interval for the population mean (μ) using the t-distribution. The t-distribution is used when the population standard deviation is unknown, and the sample standard deviation is used instead.
3. Step 3: Recall the formula for degrees of freedom when using the t-distribution. For a single sample, the degrees of freedom are calculated as: $\mathrm{df}=n-\; 1$, where n is the sample size.
4. Step 4: Locate the sample size (n) from the problem or the referenced Exercise 7 'Requirements.' Subtract 1 from the sample size to determine the degrees of freedom. For example, if the sample size is 10, the degrees of freedom would be: $\mathrm{df}=\; 10\; -\; 1\; =\; 9$.
5. Step 5: Use the calculated degrees of freedom (df) in the t-distribution table or software to find the critical t-value for constructing the confidence interval. This value depends on the desired confidence level (e.g., 95%) and the degrees of freedom.

Key Concepts

Degrees of Freedom

Degrees of freedom (df) refer to the number of independent values or quantities that can vary in a statistical calculation. In the context of estimating parameters, it often represents the number of observations minus the number of parameters being estimated. For example, in a sample of size n, when estimating the mean, the degrees of freedom would be n-1.

t Distribution

The t distribution is a type of probability distribution that is symmetric and bell-shaped, similar to the normal distribution but with heavier tails. It is used primarily in hypothesis testing and constructing confidence intervals when the sample size is small and the population standard deviation is unknown. The shape of the t distribution changes with the degrees of freedom, becoming closer to the normal distribution as df increases.

Confidence Interval

A confidence interval is a range of values, derived from sample statistics, that is likely to contain the true population parameter with a specified level of confidence, typically expressed as a percentage (e.g., 95%). It provides an estimate of uncertainty around the sample mean and is calculated using the sample mean, the t value from the t distribution, and the standard error of the mean. The width of the interval reflects the variability in the data and the sample size.