c. Find the t-value such that the area left of the t-value is 0.01 with 18 degrees of freedom. (Hint: Use symmetry.)
d. Find the critical t-value that corresponds to 90% confidence. Assume 20 degrees of freedom.

Verified step by step guidance

1. For part (c), understand that the problem asks for a t-value where the cumulative area to the left is 0.01 with 18 degrees of freedom. This means you are looking for the 1st percentile point of the t-distribution with 18 degrees of freedom.
2. Use the symmetry property of the t-distribution: since it is symmetric about zero, the t-value with area 0.01 to the left is the negative of the t-value with area 0.99 to the left. So, find the t-value corresponding to 0.99 area to the left and then take its negative.
3. For part (d), a 90% confidence level means the middle 90% of the distribution is covered, leaving 5% in each tail. So, you need to find the t-value that cuts off the upper 5% (or equivalently, has 95% area to the left) with 20 degrees of freedom.
4. Look up the t-distribution table or use statistical software to find the t-value for 0.99 area to the left with 18 degrees of freedom (for part c) and the t-value for 0.95 area to the left with 20 degrees of freedom (for part d).
5. Remember that the critical t-value for a confidence interval is positive and corresponds to the upper tail cutoff, so for part (d), the critical t-value is the positive t-value with 5% in the upper tail.

Key Concepts

t-Distribution

The t-distribution is a probability distribution used when estimating population parameters when the sample size is small and the population standard deviation is unknown. It is similar to the normal distribution but has heavier tails, which account for the extra uncertainty. The shape depends on degrees of freedom, affecting critical values.

Degrees of Freedom

Degrees of freedom (df) refer to the number of independent values that can vary in an analysis without breaking any constraints. In t-distributions, df typically equal the sample size minus one. The value of df influences the shape of the t-distribution and the critical t-values used in hypothesis testing and confidence intervals.

Critical t-Value and Confidence Intervals

A critical t-value marks the cutoff point(s) on the t-distribution for a given confidence level or tail probability. For confidence intervals, it defines the range within which the true population parameter lies with a specified probability. Finding this value involves using the area under the curve and degrees of freedom, often requiring symmetry properties of the t-distribution.