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When we talk about degrees of freedom in statistics, we're referring to the number of independent values that can vary in an analysis without breaking any constraints. In the context of a t-distribution which is used when dealing with small sample sizes or when the population standard deviation is unknown, degrees of freedom are crucial for determining the exact shape of the distribution.

The formula for calculating degrees of freedom in the case of a single sample t-test is quite straightforward: it's simply the sample size (\( n \)) minus one (\( n-1 \)). The reason we subtract one is that the sample mean is used as an estimate of the population mean, which imposes one condition on the data set, thus reducing the number of free values by one.

In the exercise at hand, with a sample size of 20, we have 19 degrees of freedom (\( 20 - 1 = 19 \)). The degrees of freedom directly impact the tails of the t-distribution — the fewer the degrees of freedom, the fatter the tails. This means that with a smaller sample size, there's more uncertainty, which the t-distribution accounts for.

Sample size plays a pivotal role in statistical analysis and inference. It determines how well the sample represents the population from which it is drawn and impacts the precision of statistical estimates. In general, larger sample sizes reduce the margin of error and lead to more reliable results.

In the presented exercise, the sample size is 20 (\(n=20\)). This number is important for two primary reasons. First, it directly informs the degrees of freedom for the t-distribution, as discussed earlier. Second, the sample size affects the standard error of the mean, which measures the dispersion of sample means around the population mean. Smaller samples result in larger standard errors, which is why when the sample size is less than 30, it's recommended to use the t-distribution rather than the standard normal distribution (z-distribution) to account for this additional uncertainty.

The term cumulative probability refers to the probability that a random variable takes on a value less than or equal to a certain number. It's a way to express how likely it is that a certain event will occur. This is calculated as the area under the probability distribution curve up to that number. For a t-distribution, which is symmetrical and bell-shaped like the normal distribution, cumulative probability is found by looking at the area under the curve to the left of a specified t-value.

In our particular exercise, we are interested in finding the cumulative probability below -1.0. t-distribution tables or software would give us the area to the right of the positive t-value (since they provide the cumulative probability from the mean to positive infinity). Therefore, to find the area below -1.0, we take the cumulative probability from the table for +1.0 and subtract it from 1. This reflects the property of the t-distribution being symmetrical around the mean.