Q: A population has mean $$\mu=100$$ and standard deviation $$\sigma=15. $$What is the standard error of the mean for samples of size $$n=25$$?

$$\text{SE} = \frac{15}{\sqrt{25}} = 3$$

Q: A normal distribution has $$\mu=60$$ and $$\sigma=8. $$What proportion of values are greater than $$x=76$$?

$$z = \frac{76-60}{8} = 2$$, \text{so} \ P(X\u003e76) = 1 - P(Z\u003c2) \approx 0.0228$

Question 1

A random variable $$X$$ has the following probability distribution: $$P(X=0)=0.2$$, $$P(X=1)=0.5$$, $$P(X=2)=0.3. $$What is the expected value $$E(X)$$?

Accepted Answer

$$E(X) = 1.1$$

Question 2

Which of the following best describes the Central Limit Theorem?

Accepted Answer

The distribution of sample means approaches a normal distribution as sample size increases, regardless of the population distribution.

Question 3

A binomial random variable $$X$$ has $$n=10$$ trials and $$p=0.3. $$What is the probability of getting exactly 4 successes?

Accepted Answer

$$P(X=4) = \frac{10!}{4!6!} \cdot 0.3^4$ \cdot 0.7^6$

Question 4

If a normal distribution has mean $$\mu=50$$ and standard deviation $$\sigma=10$$, what is the $$z-$$score for $$x=65$$?

Accepted Answer

$$z = \frac{65-50}{10} = 1.5$$

Question 5

A fair die is rolled 6 times. What is the probability that exactly two rolls show a 4?

Accepted Answer

$$P = \binom{6}{2} \cdot \left(\frac{1}{6}\$$right)^2$$ \cdot \left(\frac{5}{6}\$$right)^4$$

Question 6

Which of the following is NOT a property of a normal distribution?

Accepted Answer

The distribution is skewed to the right.

Question 7

A population has mean $$\mu=100$$ and standard deviation $$\sigma=15. $$What is the standard error of the mean for samples of size $$n=25$$?

Accepted Answer

$$	ext{SE} = \frac{15}{\sqrt{25}} = 3$$

Question 8

If $$X is a $$binomial random variable with $$n=8$$ and $$p=0.5$$, what is the expected value $$E(X)$$?

Accepted Answer

$$E(X) = n \cdot p = 8 \cdot 0.5 = 4$$

Question 9

A boxplot is used to display which of the following?

Accepted Answer

The five-number summary: minimum, Q1, median, Q3, maximum.

Question 10

A normal distribution has $$\mu=60$$ and $$\sigma=8. $$What proportion of values are greater than $$x=76$$?

Accepted Answer

$$z = \frac{76-60}{8} = 2$$, 	ext{so} \ P(X\u003e76) = 1 - P(Z\u003c2) \approx 0.0228$

Question 11

Which statement about the sampling distribution of the sample mean is correct?

Accepted Answer

It is approximately normal if the sample size is large, regardless of the population distribution.

Question 12

If a random variable $$X$$ has mean $$\mu=20$$ and variance $$\$$sigma^2$$=9$$, what is the standard deviation?

Accepted Answer

$$\sigma = 3$$

Question 13

A normal distribution has mean $$\mu=0$$ and standard deviation $$\sigma=1. $$What is the probability that $$Z is $$between $$-1$$ and $$1$$?

Accepted Answer

$$P(-1 \u003c Z \u003c 1) \approx 0.6826$$

Question 14

Which of the following is a discrete probability distribution?

Accepted Answer

Binomial distribution

Question 15

If the probability of success in a single trial is $$p=0.4$$, what is the probability of failure?

Accepted Answer

$$1-p = 0.6$$

Question 16

A sample of 100 students has a mean score of 75 and a standard deviation of 10. What is the $$z-$$score for a student who scored 90?

Accepted Answer

$$z = \frac{90-75}{10} = 1.5$$

Question 17

Which of the following is true about the area under the normal curve?

Accepted Answer

The total area under the curve is 1.

Question 18

If a random variable $$X$$ has a probability distribution such that $$P(X=1)=0.3$$, $$P(X=2)=0.4$$, $$P(X=3)=0.3$$, what is $$P(X\u003c2)$$?

Accepted Answer

$$P(X\u003c2) = 0.3$$

Guided Study Notes for Probability, Normal, and Sampling Distributions

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