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This question involves finding areas under the t-distribution curve for a given number of degrees of freedom (in this case, 16 degrees of freedom). These areas correspond to probabilities.

To solve these problems, we need to use the **t-distribution table** or a **statistical calculator** that provides cumulative probabilities for a t-distribution with 16 degrees of freedom. The areas (probabilities) will depend on whether the question asks for the area to the right, left, or between certain values.

Let's go through each part one by one:

### a. To the right of 2.120
This means we need the area under the t-distribution curve from \( t = 2.120 \) to infinity.
We can find \( P(T > 2.120) \).

### b. To the left of 1.337
Here, we are asked for the cumulative probability to the left of \( t = 1.337 \), i.e., \( P(T < 1.337) \).

### c. To the left of -1.746
This asks for the area to the left of \( t = -1.746 \), i.e., \( P(T < -1.746) \).

### d. To the right of 2.583
We need the area from \( t = 2.583 \) to infinity, i.e., \( P(T > 2.583) \).

### e. Between -2.120 and 2.120
This asks for the area between \( t = -2.120 \) and \( t = 2.120 \), i.e., \( P(-2.120 < T < 2.120) \).

### f. Between -1.746 and 1.746
This asks for the probability between \( t = -1.746 \) and \( t = 1.746 \), i.e., \( P(-1.746 < T < 1.746) \).

I will calculate each of these probabilities using statistical tools.Here are the calculated probabilities for each part:

- **a. To the right of 2.120**: \( 0.025 \)
- **b. To the left of 1.337**: \( 0.900 \)
- **c. To the left of -1.746**: \( 0.050 \)
- **d. To the right of 2.583**: \( 0.010 \)
- **e. Between -2.120 and 2.120**: \( 0.950 \)
- **f. Between -1.746 and 1.746**: \( 0.900 \)

Would you like more details on any of these calculations?

Here are some related questions for practice:
1. What is the area to the left of 2.120 for 16 degrees of freedom?
2. How would the probability to the right of 1.337 change if degrees of freedom were 20?
3. Find the area between -2.583 and 2.583 for 16 degrees of freedom.
4. What is the area to the right of -1.746 for 10 degrees of freedom?
5. How does the t-distribution compare to the normal distribution as degrees of freedom increase?

**Tip:** The t-distribution is symmetric, so probabilities to the left of negative values are the same as to the right of their positive counterparts.

For a t distribution with 16 degrees of freedom, find the area, or probability, in each region: to the right of 2.120, to the left of 1.337, to the left of -1.746, to the right of 2.583, between -2.120 and 2.120, between -1.746 and 1.746.

Explore how to calculate probabilities for different areas under the t-distribution curve with 16 degrees of freedom. Learn about finding areas to the right and left of specific values as well as between values using the t-distribution.

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