Q: A clinical trial finds that 44 out of 100 patients experienced an adverse reaction to a drug. What is the 95% confidence interval for the proportion of adverse reactions?

$$0.44 \pm 1.96 \sqrt{\dfrac{0.44 \times 0.56}{100}}$$

Q: Suppose you want to estimate a population proportion with a 95% confidence interval and a margin of error no greater than 0.03. What is the minimum sample size required if $$p is $$unknown?

$$n = \dfrac{(1.96)^2$ \times 0.25}{(0.03)^2$}$$

Question 1

Which formula is used to calculate the 95% confidence interval estimate for a population proportion $$p$$ when the sample proportion is $$\hat{p}$$ and sample size is $$n$$?

Accepted Answer

$$\hat{p} \pm 1.96 \sqrt{\dfrac{\hat{p}(1-\hat{p})}{n}}$$

Question 2

A clinical trial finds that 44 out of 100 patients experienced an adverse reaction to a drug. What is the 95% confidence interval for the proportion of adverse reactions?

Accepted Answer

$$0.44 \pm 1.96 \sqrt{\dfrac{0.44 	imes 0.56}{100}}$$

Question 3

When the population proportion $$p is $$unknown, what conservative value is often used for $$p in $$sample size calculations?

Accepted Answer

$$p = 0.50$$

Question 4

Which of the following best describes the margin of error (ME) in the context of estimating a population proportion?

Accepted Answer

The maximum likely difference between the observed sample proportion and the true population proportion

Question 5

Suppose you want to estimate a population proportion with a 95% confidence interval and a margin of error no greater than 0.03. What is the minimum sample size required if $$p is $$unknown?

Accepted Answer

$$n = \dfrac{(1.96)^2$ 	imes 0.25}{(0.03)^2$}$$

Question 6

A researcher wants to estimate the mean systolic blood pressure in a population. The sample mean is 130 mmHg, the sample standard deviation is 15 mmHg, and the sample size is 36. What is the 95% confidence interval for the population mean?

Accepted Answer

$$130 \pm 1.96 \dfrac{15}{\sqrt{36}}$$

Question 7

Which statement is TRUE regarding the use of the $$t-$$distribution in confidence interval estimation?

Accepted Answer

The $$t-$$distribution is used when the population standard deviation is unknown and the sample size is small

Question 8

What is the formula for the margin of error (ME) when estimating a population mean with unknown variance and sample size $$n$$?

Accepted Answer

$$ME = t_{\alpha/2, df} \dfrac{s}{\sqrt{n}}$$

Question 9

A sample of 10 bumblebee hat sizes has a sample variance $$s^2 = 1.14$. What is the 95% confidence interval for the population variance?

Accepted Answer

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

Question 10

Which of the following is the best point estimate of the population mean $$\mu$$?

Accepted Answer

The sample mean $$\bar{x}$$

Question 11

In the context of confidence intervals, what does a 95% confidence level mean?

Accepted Answer

If we repeated the sampling process many times, about 95% of the intervals would contain the true population parameter

Question 12

A physician is concerned about cardiac deaths favored by manual snow shoveling. What single value in the confidence interval from part (a) would be of greatest concern?

Accepted Answer

The upper limit of the confidence interval

Question 13

Which formula is used to estimate the required sample size $$n$$ for a desired margin of error $$E$$ when estimating a population mean with known standard deviation $$\sigma$$?

Accepted Answer

$$n = \left( \dfrac{z_{\alpha/2} \sigma}{E} \$$right)^2$$

Question 14

What is the number of degrees of freedom for a sample of size $$n$$ when estimating the population variance?

Accepted Answer

$$n - 1$$

Question 15

Which of the following is NOT a requirement for using the normal approximation to estimate a population proportion?

Accepted Answer

The sample size $$n$$ must be less than 30

Question 16

A study finds 122 correct identifications and 157 wrong identifications in 280 trials. What is the sample proportion of correct identifications?

Accepted Answer

$$\dfrac{122}{280}$$

Question 17

Which formula is used to construct a confidence interval for the population variance $$\$$sigma^2$$ with a normal sample?

Accepted Answer

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

Question 18

If the value of a parameter $$p is $$outside the interval $$(0.32, 0.68)$$, what is the probability that $$p is $$inside the interval?

Accepted Answer

$$0$$

Estimating Parameters and Determining Sample Sizes: Study Notes

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