Q: What is the standard deviation of a binomial distribution with $$n=20$$ trials and probability of success $$p=0.3$$?

$$\sigma = \sqrt{np(1-p)} = \sqrt{20 \times 0.3 \times 0.7} \approx 2.05$$

Question 1

Which of the following is a key difference between discrete and continuous random variables?

Accepted Answer

Discrete random variables have countable values, while continuous random variables have measurable values.

Question 2

Which of the following is NOT a requirement for a binomial probability distribution?

Accepted Answer

The probability of success changes from trial to trial.

Question 3

If a fair coin is flipped 10 times, what is the probability of getting exactly 6 heads? Use the binomial formula $$P(X=k) = \frac{n!}{k!(n-k)!} p^k (1-p$$)^{n-k}$$ where n=10$, k=6$, and p=0.5$.

Accepted Answer

$$P(X=6) = 0.205$$

Question 4

Which formula is used to find the mean (expected value) of a binomial distribution?

Accepted Answer

$$\mu = np$$

Question 5

A normal distribution is described as:

Accepted Answer

Bell-shaped and symmetric

Question 6

What is the standard deviation of a binomial distribution with $$n=20$$ trials and probability of success $$p=0.3$$?

Accepted Answer

$$\sigma = \sqrt{np(1-p)} = \sqrt{20 	imes 0.3 	imes 0.7} \approx 2.05$$

Question 7

If a population is normally distributed with mean $$\mu = 100$$ and standard deviation $$\sigma = 15$$, what is the $$z-$$score for a value of $$x = 130$$?

Accepted Answer

$$z = \frac{130-100}{15} = 2$$

Question 8

Which rule is commonly used to identify unusual values in a normal distribution?

Accepted Answer

Range Rule of Thumb

Question 9

A sample of 40 students has a mean test score of 75 and a standard deviation of 10. What is the standard error of the mean?

Accepted Answer

$$SE = \frac{10}{\sqrt{40}} \approx 1.58$$

Question 10

Which of the following is an unbiased estimator of a population mean?

Accepted Answer

Sample mean

Question 11

When is it appropriate to use the Central Limit Theorem to approximate probabilities for sample means?

Accepted Answer

When the sample size is large (usually $$n \geq 30)$$

Question 12

A researcher wants to estimate the proportion of people who prefer tea over coffee. Which formula should be used to construct a confidence interval for the population proportion?

Accepted Answer

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

Question 13

Which graphical method is best for assessing the symmetry and spread of a data set?

Accepted Answer

Boxplot

Question 14

A continuous uniform distribution has which of the following properties?

Accepted Answer

All values within the range are equally probable.

Question 15

If a sample mean is 50, the population mean is 52, and the standard error is 2, what is the $$z-$$score for the sample mean?

Accepted Answer

$$z = \frac{50-52}{2} = -1$$

Question 16

Which of the following statements about unbiased estimators is true?

Accepted Answer

An unbiased estimator's expected value equals the population parameter.

Question 17

To estimate a population mean, which formula is used to determine the required sample size $$n$$ for a given margin of error $$E$$ and confidence level $$z^*$$?

Accepted Answer

$$n = \left( \frac{z^* \sigma}{E} \$$right)^2$$

Question 18

Which of the following is NOT a valid method for estimating binomial probabilities?

Accepted Answer

Using the mean and standard deviation only

Question 19

A sample of 25 patients has a mean cholesterol level of 190 mg/dL and a sample standard deviation of 20 mg/dL. What is the 95% confidence interval for the population mean? (Use $$t^* = 2.064$$ for $$df=24)$$

Accepted Answer

$$190 \pm 2.064 	imes \frac{20}{\sqrt{25}} = 190 \pm 8.256$$

Question 20

Which of the following best describes the Central Limit Theorem?

Accepted Answer

The sampling distribution of the sample mean approaches a normal distribution as the sample size increases, regardless of the population's distribution.

Study Guide: Random Variables, Binomial Distributions, and Statistical Inference (Chapters 5–7)

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