The image asks for the area or probability in specific regions of a t-distribution with 16 degrees of freedom. These are calculated using the t-distribution table or using statistical software. I'll calculate each one and round them according to the requirements:

Given: t-distribution with 16 degrees of freedom.

a. To the right of 2.120

We need $P(T > 2.120)$. Using a t-distribution table or calculator: $P(T > 2.120) = 0.025$

b. To the left of 1.337

We need $P(T < 1.337)$. Using a t-distribution table or calculator: $P(T < 1.337) = 0.900$

c. To the left of -1.746

We need $P(T < -1.746)$. Since the t-distribution is symmetric, $P(T < -1.746) = P(T > 1.746)$: $P(T < -1.746) = 0.05$

d. To the right of 2.583

We need $P(T > 2.583)$. Using a t-distribution table or calculator: $P(T > 2.583) = 0.01$

e. Between -2.120 and 2.120

We need $P(-2.120 < T < 2.120)$. This corresponds to $1 - 2P(T > 2.120)$: $P(-2.120 < T < 2.120) = 1 - 2(0.025) = 0.950$

f. Between -1.746 and 1.746

We need $P(-1.746 < T < 1.746)$. This corresponds to $1 - 2P(T > 1.746)$: $P(-1.746 < T < 1.746) = 1 - 2(0.05) = 0.900$

Summary:

• a: $P(T > 2.120) = 0.025$ (to 3 decimals)
• b: $P(T < 1.337) = 0.90$ (to 2 decimals)
• c: $P(T < -1.746) = 0.05$ (to 2 decimals)
• d: $P(T > 2.583) = 0.01$ (to 2 decimals)
• e: $P(-2.120 < T < 2.120) = 0.95$ (to 2 decimals)
• f: $P(-1.746 < T < 1.746) = 0.90$ (to 2 decimals)

Would you like further details or clarifications on any of these calculations?

Relative Questions:

1. How do we use a t-distribution table for different degrees of freedom?
2. What is the symmetry property of the t-distribution?
3. How does changing degrees of freedom affect the t