Q: A confidence interval for a population proportion is calculated as $$\hat{p} \pm z^* \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}. If$$ $$\hat{p} = 0.4$$, $$n = 200$$, and $$z^* = 1.96$$, what is the margin of error?

$$1.96 \sqrt{\frac{0.4 \times 0.6}{200}}$$

Question 1

Which of the following best describes the shape of a normal probability distribution?

Accepted Answer

It is symmetric and bell-shaped.

Question 2

A hospital records the systolic blood pressure of 10,000 patients. If the distribution is normal with mean $$\mu = 120$$ and standard deviation $$\sigma = 15$$, what percentage of patients have systolic blood pressure between 105 and 135?

Accepted Answer

About 68%

Question 3

Which formula is used to standardize a value $$x in a $$normal distribution?

Accepted Answer

$$z = \frac{x - \mu}{\sigma}$$

Question 4

A sample of 50 students has a mean test score of 80 and a population standard deviation of 10. What is the standard error of the mean?

Accepted Answer

$$\frac{10}{\sqrt{50}}$$

Question 5

Which statement about sampling distributions is correct?

Accepted Answer

The sampling distribution of the sample mean approaches normality as sample size increases, regardless of population shape.

Question 6

A pharmaceutical company wants to estimate the mean recovery time for a new drug. Which of the following is a point estimate for the population mean?

Accepted Answer

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

Question 7

If a confidence interval for the mean is calculated as $$\bar{x} \pm z^* \frac{\sigma}{\sqrt{n}}$$, what does $$z^*$$ represent?

Accepted Answer

The critical value from the standard normal distribution for the desired confidence level.

Question 8

A researcher wants to estimate the proportion of voters who support a new policy. Which formula should be used to calculate the margin of error for the proportion?

Accepted Answer

$$z^* \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$$

Question 9

Which of the following is NOT an assumption for constructing a confidence interval for a population mean?

Accepted Answer

The sample size is less than 30.

Question 10

A scientist collects a random sample of 100 water samples and finds a mean lead concentration of 5 ppm with a standard deviation of 2 ppm. What is the 95% confidence interval for the mean lead concentration?

Accepted Answer

$$5 \pm 1.96 \frac{2}{\sqrt{100}}$$

Question 11

Which of the following best explains why larger sample sizes yield narrower confidence intervals?

Accepted Answer

Larger sample sizes decrease the standard error, making the estimate more precise.

Question 12

A population is not normally distributed. According to the Central Limit Theorem, what happens to the sampling distribution of the sample mean as sample size increases?

Accepted Answer

It becomes approximately normal.

Question 13

Which of the following is a parameter, not a statistic?

Accepted Answer

Population mean $$\mu$$

Question 14

A confidence interval for a population proportion is calculated as $$\hat{p} \pm z^* \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}. If$$ $$\hat{p} = 0.4$$, $$n = 200$$, and $$z^* = 1.96$$, what is the margin of error?

Accepted Answer

$$1.96 \sqrt{\frac{0.4 	imes 0.6}{200}}$$

Question 15

Which scenario best illustrates the use of a normal probability distribution in real life?

Accepted Answer

Modeling the distribution of heights in a population.

Question 16

If a sample mean is used to estimate a population mean, what is the term for this estimate?

Accepted Answer

Point estimate

Question 17

Which of the following is true about the mean and standard deviation of the sampling distribution of the sample mean?

Accepted Answer

Mean is $$\mu$$, standard deviation is $$\frac{\sigma}{\sqrt{n}}$$

Question 18

A confidence interval for the mean is wider when:

Accepted Answer

The confidence level is higher.

Question 19

Which formula represents the probability density function of the normal distribution?

Accepted Answer

$$f(x) = \frac{1}{\sigma \sqrt{2\pi}} e$$^{-\frac{1}$${2} \left( \frac{x-\mu}{\sigma} \$$right)^2$$}$$

Question 20

A researcher wants to estimate the mean weight of newborns in a city. Which of the following would increase the precision of the estimate?

Accepted Answer

Increasing the sample size.

Study Notes: The Normal Probability Distribution, Sampling Distributions, and Estimating the Value of a Parameter

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