Q: A sample of $$n = 25$$ has a mean $$\bar{x} = 80$$ and standard deviation $$s = 10. $$What is the 90% confidence interval for the population mean?

The interval is $$\bar{x} \pm t_{0.05,24} \times \frac{s}{\sqrt{n}}$$. $$t_{0.05,24} \approx 1.711. So$$, $$80 \pm 1.711 \times \frac{10}{5} = 80 \pm 3.422. $$Interval: $$(76.578, 83.422)$$

Q: A hospital wants to estimate the mean recovery time for a new treatment. If they want a margin of error of $$\pm 2$$ days with 95% confidence and the population standard deviation is estimated at $$\sigma = 8$$ days, what sample size is needed?

Use $$n = \left(\frac{z_{0.025} \times \sigma}{E}\$$right)^2$$. z$$_{0.025} = 1.96$$, E = 2$. n$$ = \left(\frac{1.96 \times 8}{2}\$$right)^2$$ = $$(7.84)^2 = 61.47$. Round up: n = 62$

Q: A pharmaceutical company tests a new drug on a sample of $$n = 100$$ patients. The sample mean recovery time is $$\bar{x} = 12$$ days, and the population standard deviation is $$\sigma = 4$$ days. What is the 95% confidence interval for the mean recovery time?

The interval is $$\bar{x} \pm z_{0.025} \times \frac{\sigma}{\sqrt{n}}$$. $$z_{0.025} = 1.96$$. $$12 \pm 1.96 \times \frac{4}{10} = 12 \pm 0.784. $$Interval: $$(11.216, 12.784)$$

Question 1

Which of the following statements best describes the Central Limit Theorem (CLT)?

Accepted Answer

The CLT states that the distribution of sample means approaches a normal distribution as the sample size increases, regardless of the population's distribution.

Question 2

A researcher wants to estimate the mean blood pressure of adults in a city. Which condition must be met to use the normal distribution for the distribution of sample means?

Accepted Answer

The sample size must be large (typically $$n \geq 30) or $$the population must be normally distributed.

Question 3

Suppose the population mean IQ is $$\mu = 100$$ with $$\sigma = 15. If a $$random sample of $$n = 36 is $$taken, what is the standard error of the mean?

Accepted Answer

The standard error is $$\frac{\sigma}{\sqrt{n}} = \frac{15}{\sqrt{36}} = 2.5$$

Question 4

A population has mean $$\mu = 50$$ and standard deviation $$\sigma = 10. $$What is the probability that a sample mean from $$n = 64$$ falls between 48 and 52?

Accepted Answer

First, calculate the standard error: $$\frac{10}{\sqrt{64}} = 1.25. $$Then, $$z-$$scores: $$z_1$$ = \frac{48-50}{1.25} = -1.6$$, z_2$ = \frac{52-50}{1.25} = 1.6. $$Using the normal table, $$P(-1.6 \u003c z \u003c 1.6) \approx 0.8904 - 0.0548 = 0.8356$$

Question 5

Which of the following is NOT a property of confidence intervals for population means?

Accepted Answer

The confidence interval always contains the population mean.

Question 6

A 95% confidence interval for the mean weight of newborns is $$(3.2, 3.8) kg. $$What does this mean?

Accepted Answer

We are 95% confident that the true mean weight of newborns is between 3.2 kg and 3.8 kg.

Question 7

A sample of $$n = 25$$ has a mean $$\bar{x} = 80$$ and standard deviation $$s = 10. $$What is the 90% confidence interval for the population mean?

Accepted Answer

The interval is $$\bar{x} \pm t_{0.05,24} 	imes \frac{s}{\sqrt{n}}$$. $$t_{0.05,24} \approx 1.711. So$$, $$80 \pm 1.711 	imes \frac{10}{5} = 80 \pm 3.422. $$Interval: $$(76.578, 83.422)$$

Question 8

Which factor will NOT increase the width of a confidence interval for a population mean?

Accepted Answer

Increasing the sample size.

Question 9

A hospital wants to estimate the mean recovery time for a new treatment. If they want a margin of error of $$\pm 2$$ days with 95% confidence and the population standard deviation is estimated at $$\sigma = 8$$ days, what sample size is needed?

Accepted Answer

Use $$n = \left(\frac{z_{0.025} 	imes \sigma}{E}\$$right)^2$$. z$$_{0.025} = 1.96$$, E = 2$. n$$ = \left(\frac{1.96 	imes 8}{2}\$$right)^2$$ = $$(7.84)^2 = 61.47$. Round up: n = 62$

Question 10

If the population is not normally distributed, which sample size is generally considered sufficient for the Central Limit Theorem to apply?

Accepted Answer

A sample size of $$n \geq 30 is $$generally considered sufficient.

Question 11

A pharmaceutical company tests a new drug on a sample of $$n = 100$$ patients. The sample mean recovery time is $$\bar{x} = 12$$ days, and the population standard deviation is $$\sigma = 4$$ days. What is the 95% confidence interval for the mean recovery time?

Accepted Answer

The interval is $$\bar{x} \pm z_{0.025} 	imes \frac{\sigma}{\sqrt{n}}$$. $$z_{0.025} = 1.96$$. $$12 \pm 1.96 	imes \frac{4}{10} = 12 \pm 0.784. $$Interval: $$(11.216, 12.784)$$

Question 12

Which of the following best explains why increasing the sample size reduces the margin of error in a confidence interval?

Accepted Answer

Because the standard error $$\frac{\sigma}{\sqrt{n}}$$ decreases as $$n$$ increases.

Question 13

A random sample of $$n = 49$$ yields a sample mean $$\bar{x} = 20$$ and population standard deviation $$\sigma = 7. $$What is the probability that the sample mean is less than 18?

Accepted Answer

Standard error: $$\frac{7}{\sqrt{49}} = 1$$. $$z = \frac{18-20}{1} = -2$$. $$P(z \u003c -2) \approx 0.0228$$

Question 14

Which scenario is an appropriate real-life application of the Central Limit Theorem?

Accepted Answer

A quality control manager at a factory uses the average weight of boxes from random samples to assess whether the production process is consistent.

Question 15

If a confidence interval for a population mean is $$(45, 55)$$, which of the following statements is correct?

Accepted Answer

We are confident that the population mean lies between 45 and 55.

Question 16

A sample of $$n = 16$$ has a mean $$\bar{x} = 30$$ and standard deviation $$s = 8. $$What is the 99% confidence interval for the population mean?

Accepted Answer

The interval is $$\bar{x} \pm t_{0.005,15} 	imes \frac{s}{\sqrt{n}}$$. $$t_{0.005,15} \approx 2.947$$. $$30 \pm 2.947 	imes 2 = 30 \pm 5.894. $$Interval: $$(24.106, 35.894)$$

Question 17

Which of the following is a correct formula for constructing a confidence interval for a population mean when the population standard deviation is unknown?

Accepted Answer

The confidence interval is $$\bar{x} \pm t_{\alpha/2, n-1} 	imes \frac{s}{\sqrt{n}}$$

Question 18

A sample of $$n = 50$$ has a mean $$\bar{x} = 100$$ and population standard deviation $$\sigma = 20. $$What is the margin of error for a 95% confidence interval?

Accepted Answer

Margin of error: $$z_{0.025} 	imes \frac{\sigma}{\sqrt{n}} = 1.96 	imes \frac{20}{\sqrt{50}} = 1.96 	imes 2.828 = 5.544$$

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