Q: A pharmaceutical company wants to estimate the mean effect of a new drug. They want the margin of error to be no more than 2 units, with 95% confidence. The population standard deviation is estimated to be 10. What is the minimum sample size required?

$$n = \left(\dfrac{1.96 \times 10}{2}\$$right)^2$$

Question 1

A researcher wants to estimate the mean blood pressure of a population. The population standard deviation $$\sigma is $$unknown, and a sample of $$n=18 is $$taken. Which distribution should be used to construct a 95% confidence interval for the mean?

Accepted Answer

The Student's t-distribution with 17 degrees of freedom

Question 2

Which of the following is a property of the Student's t-distribution?

Accepted Answer

It is centered at 0 and is symmetric about 0

Question 3

Suppose you have a sample of $$n=25$$ from a normal population with unknown mean and unknown standard deviation. The sample mean is $$\overline{x}=50$$ and the sample standard deviation is $$s=8. $$What is the correct formula for a 95% confidence interval for the population mean?

Accepted Answer

$$\overline{x} \pm t_{0.025,24} \dfrac{s}{\sqrt{n}}$$

Question 4

Which of the following statements about the tails of the t-distribution is correct?

Accepted Answer

The t-distribution has fatter tails than the standard normal distribution

Question 5

A pharmaceutical company wants to estimate the mean effect of a new drug. They want the margin of error to be no more than 2 units, with 95% confidence. The population standard deviation is estimated to be 10. What is the minimum sample size required?

Accepted Answer

$$n = \left(\dfrac{1.96 	imes 10}{2}\$$right)^2$$

Question 6

Which of the following is NOT a characteristic of the chi-square distribution?

Accepted Answer

It is centered at zero and symmetric about zero

Question 7

A sample of $$n=12$$ car prices has a sample standard deviation of $$s=2615.19. To $$construct a 90% confidence interval for the population variance, which formula should be used for the lower bound?

Accepted Answer

$$\dfrac{(n-1)s^2$}{\$$chi^2$$_{\alpha/2}}$$

Question 8

Why is the area in the tails of the t-distribution greater than that of the standard normal distribution?

Accepted Answer

Because the t-distribution uses the sample standard deviation as an estimate of the population standard deviation, introducing more variability

Question 9

A researcher collects a sample of $$n=40$$ from a population with unknown mean and unknown standard deviation. Which distribution should be used to construct a confidence interval for the mean?

Accepted Answer

The standard normal (z) distribution

Question 10

Which of the following is a correct interpretation of a 95% confidence interval for a population mean?

Accepted Answer

If we were to take many samples and construct a confidence interval from each, about 95% of those intervals would contain the true population mean

Question 11

Which of the following conditions must be met to use the chi-square distribution to construct a confidence interval for the population variance?

Accepted Answer

The data must come from a normal distribution

Question 12

A sample of $$n=20 is $$drawn from a normal population with unknown mean and unknown standard deviation. The sample mean is $$\overline{x}=12.5$$ and $$s=5. $$What is the 95% confidence interval for the mean if $$t_{0.025,19}=2.093$$?

Accepted Answer

$$(12.5 \pm 2.093 	imes \dfrac{5}{\sqrt{20}})$$

Question 13

Which of the following best describes the effect of increasing sample size on the t-distribution?

Accepted Answer

The t-distribution approaches the standard normal distribution

Question 14

A 90% confidence interval for the variance $$\$$sigma^2$$ of a normal population is constructed using a sample of size n=12$. The sample variance is s^2$. Which critical values are used in the formula for the interval?

Accepted Answer

$$\$$chi^2$$_{0.95,11}$$ and $$\$$chi^2$$_{0.05,11}$$

Question 15

Which of the following is NOT a correct step when checking if it is appropriate to use the t-distribution for a small sample ($$n\u003c30$$)?

Accepted Answer

Use the t-distribution regardless of skewness or outliers

Question 16

A 98% confidence interval for the mean of a normal population is wider than a 95% confidence interval using the same data. Why?

Accepted Answer

Because the critical value $$t_{\alpha/2,df} is $$larger for 98% confidence

Question 17

Which formula is used to estimate the sample standard deviation $$s$$ from a sample of $$n$$ observations $$X_1$$, $$X_2$$, ..., X_n$$?

Accepted Answer

$$s = \sqrt{\dfrac{1}{n-1} \sum_{i=1}^n (X_i - \overline{X})^2$}$$

Question 18

A confidence interval for the population variance is not of the form 'point estimate ± margin of error' because:

Accepted Answer

The sampling distribution of the sample variance is not symmetric

Lecture 27: Confidence Intervals for Mean, Variance, and Standard Deviation (Student's t and Chi-Square Distributions)

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