A researcher conducts a hypothesis test using a sample of n=20 feom an unknown population What are the degrees of freedom
The correct answer and explanation is:
The degrees of freedom for a sample from an unknown population, when conducting a hypothesis test, are typically calculated as $n – 1$. Here, $n$ represents the sample size. In this case, since the sample size is given as $n = 20$, the degrees of freedom would be:
$$
\text{Degrees of freedom} = n – 1 = 20 – 1 = 19
$$
The concept of degrees of freedom is tied to the number of independent pieces of information available in a sample. In hypothesis testing, especially when estimating population parameters (like the mean) using sample data, the degrees of freedom reflect how many values are free to vary once certain parameters have been estimated.
For instance, when calculating the sample variance or standard deviation, we use the sample mean, which has already been calculated from the data. Since the sample mean is an estimate of the population mean, once you know 19 of the data points, the last one is constrained. Therefore, only 19 pieces of data are “free” to vary in this case. The remaining piece of data is dependent on the others, given that the sum of deviations from the sample mean must always equal zero.
In hypothesis testing, the degrees of freedom also play a crucial role in determining the appropriate distribution for the test statistic. For example, if conducting a t-test, the t-distribution with 19 degrees of freedom would be used to find the critical value and calculate the p-value.
In summary, the degrees of freedom for a sample of size 20 is 19, calculated as $n – 1$. This adjustment accounts for the fact that one parameter (the sample mean) has already been estimated from the data, thereby reducing the number of independent values.