Q: A sample of 30 babies has a mean birthweight of $$\bar{y} = 3498.7$$ grams and $$s = 434.2$$ grams. What is the 90% confidence interval for the mean birthweight? ($$t^*_{29} = 1.699)$$

$$(3498.7 \pm 1.699 \times \frac{434.2}{\sqrt{30}})$$

Question 1

Which of the following best describes what a confidence interval represents?

Accepted Answer

The likely values of the population parameter based on a sample from the population.

Question 2

A researcher collects a random sample of 138 students and finds a sample mean reading time of $$\bar{y} = 14.7$$ minutes. The sample standard deviation is $$s = 16.5$$ minutes. What is the standard error of the mean?

Accepted Answer

$$SE = \frac{16.5}{\sqrt{138}} = 1.405$$

Question 3

Which of the following is NOT an assumption or condition required for constructing a confidence interval for a mean using the t-distribution?

Accepted Answer

Population mean is known

Question 4

Why do we use the t-distribution instead of the normal (z) distribution when constructing confidence intervals for means in most practical situations?

Accepted Answer

Because the population standard deviation $$\sigma is $$unknown and we estimate it with $$s$$

Question 5

A wildlife biologist captures 10 red squirrels and finds a sample mean weight of 12.32 grams and a standard deviation of 1.88 grams. Which formula gives a 95% confidence interval for the mean weight of all squirrels?

Accepted Answer

$$12.32 \pm 2.262 \frac{1.88}{\sqrt{9}}$$

Question 6

Which of the following statements about the Student’s t-distribution is FALSE?

Accepted Answer

It has less spread than the normal distribution for small sample sizes.

Question 7

A sample of 30 babies has a mean birthweight of $$\bar{y} = 3498.7$$ grams and $$s = 434.2$$ grams. What is the 90% confidence interval for the mean birthweight? ($$t^*_{29} = 1.699)$$

Accepted Answer

$$(3498.7 \pm 1.699 	imes \frac{434.2}{\sqrt{30}})$$

Question 8

Which of the following best explains why the t-distribution has fatter tails than the normal distribution?

Accepted Answer

Because the sample standard deviation $$s is $$used to estimate $$\sigma$$, introducing more variability

Question 9

A coffee shop owner samples 10 cups and finds a mean of 12.5 ounces and a standard deviation of 0.5 ounces. Which formula gives a 95% confidence interval for the mean weight of all cups of coffee?

Accepted Answer

$$12.5 \pm 2.228 \frac{0.5}{\sqrt{10}}$$

Question 10

Which of the following is the correct interpretation of a 95% confidence interval for the mean reading time of STAT 1410 students: (11.9, 17.5) minutes?

Accepted Answer

We are 95% confident that the average time for all STAT 1410 students to read the chapter is between 11.9 and 17.5 minutes.

Question 11

Which of the following is NOT a condition for using the t-distribution to construct a confidence interval for the mean?

Accepted Answer

The population must be exactly normal, regardless of sample size.

Question 12

Which of the following statements about degrees of freedom (df) in the t-distribution is correct?

Accepted Answer

df = n - 1

Question 13

A 95% confidence interval for the mean contaminant concentration in salmon is $$(0.0834, 0.0992)$$ ppm. What does this interval mean?

Accepted Answer

We are 95% confident that the true mean contaminant concentration in all farm-raised salmon is between 0.0834 and 0.0992 ppm.

Question 14

Which of the following is TRUE about the t-distribution as the sample size increases?

Accepted Answer

It becomes more like the normal distribution.

Question 15

Which of the following formulas is used to calculate a one-sample t-interval for the mean?

Accepted Answer

$$\bar{y} \pm t^*_{n-1} \frac{s}{\sqrt{n}}$$

Question 16

Which of the following is a correct statement about the spread of the t-distribution compared to the normal distribution?

Accepted Answer

The t-distribution has more spread than the normal distribution, especially for small sample sizes.

Question 17

Which of the following is the correct formula for the margin of error (MoE) in a one-sample t-interval for the mean?

Accepted Answer

$$MoE = t^*_{n-1} 	imes \frac{s}{\sqrt{n}}$$

Question 18

Which of the following statements correctly describe t-distributions?

Accepted Answer

I and III only

Confidence Intervals for Means: The t-Distribution and Its Applications

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