Question 1

Which of the following is NOT a requirement for a function to be a probability density function (pdf) for a continuous random variable $$X$$?

Accepted Answer

The total area under the graph of the function must be less than 1.

Question 2

Suppose $$X is a $$continuous random variable uniformly distributed on the interval $$[2, 6]. $$What is the value of the probability density function $$f(x)$$ for $$2 \leq x \leq 6$$?

Accepted Answer

$$f(x) = \frac{1}{4}$$

Question 3

For a normal distribution, which of the following statements is TRUE?

Accepted Answer

The mean, median, and mode are all equal.

Question 4

If $$X is $$normally distributed with mean $$\mu = 50$$ and standard deviation $$\sigma = 8$$, what is the $$z-$$score for $$X = 66$$?

Accepted Answer

$$z = 2$$

Question 5

Which property is NOT true for the normal density curve?

Accepted Answer

It is skewed to the left.

Question 6

A sample of 36 students has a mean test score of 80 and a population standard deviation of 12. What is the standard error of the mean?

Accepted Answer

$$2$$

Question 7

According to the Central Limit Theorem, as the sample size $$n$$ increases, the sampling distribution of the sample mean:

Accepted Answer

Becomes approximately normal regardless of the population's distribution.

Question 8

Suppose a 95% confidence interval for a population proportion $$p is$$ $$(0.41, 0.53). $$What is the margin of error?

Accepted Answer

$$0.06$$

Question 9

Which of the following increases the width of a confidence interval for a population mean?

Accepted Answer

Increasing the level of confidence.

Question 10

If the sample proportion is $$\hat{p} = 0.4$$ and the sample size is $$n = 100$$, what is the standard error of $$\hat{p}$$?

Accepted Answer

$$0.049$$

Question 11

Which of the following is NOT a property of the $$t-$$distribution?

Accepted Answer

It is always narrower than the standard normal distribution.

Question 12

A researcher wants to estimate the proportion of students who own a laptop with 95% confidence and a margin of error of 0.05. If no prior estimate is available, what value of $$p$$ should be used to determine the required sample size?

Accepted Answer

$$p = 0.5$$

Question 13

Which formula gives the sample size $$n$$ needed to estimate a population mean $$\mu$$ with margin of error $$E$$ and confidence level $$(1-\alpha)$$?

Accepted Answer

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

Question 14

If the sample size increases while keeping the population standard deviation constant, what happens to the standard error of the mean?

Accepted Answer

It decreases.

Question 15

Which of the following best describes the meaning of a 95% confidence interval for a population proportion?

Accepted Answer

If we took many samples, about 95% of the intervals would contain the true population proportion.

Question 16

A population is not normally distributed. According to the Central Limit Theorem, the sampling distribution of the sample mean will be approximately normal if:

Accepted Answer

The sample size is sufficiently large.

Question 17

Which of the following is the correct formula for the standard error of the sample proportion $$\hat{p}$$?

Accepted Answer

$$\sqrt{\frac{p(1-p)}{n}}$$

Question 18

If a 99% confidence interval is wider than a 95% confidence interval for the same data, what is the main reason?

Accepted Answer

A higher confidence level requires a larger critical value.

Question 19

Which of the following is TRUE about the $$t-$$distribution compared to the standard normal distribution?

Accepted Answer

The $$t-$$distribution has heavier tails.

Continuous Random Variables, Normal Distribution, and Sampling Distributions: Study Notes

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