Question 1

Which of the following statements best describes how the t-distribution changes as the degrees of freedom increase, as shown in the image?

Accepted Answer

The t-distribution becomes more similar to the standard normal distribution.

Question 2

Based on the graph, what happens to the peak of the t-distribution as the degrees of freedom increase?

Accepted Answer

The peak becomes higher and the distribution becomes narrower.

Question 3

Suppose you are conducting a hypothesis test using the t-distribution. If your sample size increases from 5 to 30, how does the critical value for a 95% confidence interval change?

Accepted Answer

The critical value decreases and approaches the z-value for 95% confidence.

Question 4

Why is the t-distribution used instead of the normal distribution when estimating a population mean with a small sample size?

Accepted Answer

Because the t-distribution accounts for the increased variability in small samples.

Question 5

Given the t-distribution curves in the image, which curve would you use for a sample size of 12?

Accepted Answer

The curve with degrees of freedom = 10.

Question 6

If you were to overlay the standard normal distribution on the graph, how would it compare to the t-distribution with 30 degrees of freedom?

Accepted Answer

It would be almost indistinguishable from the t-distribution.

Question 7

What is the formula for the degrees of freedom when using the t-distribution for a single sample mean?

Accepted Answer

$$df = n - 1$$

Question 8

Which of the following best explains why the tails of the t-distribution are heavier for smaller degrees of freedom?

Accepted Answer

Because smaller samples have more uncertainty, increasing the probability of extreme values.

Question 9

If a researcher uses the t-distribution with 1 degree of freedom, what can be said about the shape of the distribution compared to the normal distribution?

Accepted Answer

It is less peaked and has much heavier tails.

Question 10

Which of the following is true about the area under each t-distribution curve, regardless of the degrees of freedom?

Accepted Answer

The area is always equal to 1.

Statistical Inference: Sampling Distributions, Estimation, and Hypothesis Testing

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