Q: A polygraph test for espionage returns a positive result 90% of the time if the person is guilty, and 1% of the time if the person is innocent. If 95% of suspects are actually guilty, what is the probability that a person is innocent given a positive polygraph result?

$$P(\text{Innocent} | +) = \frac{0.05 \times 0.01}{0.95 \times 0.9 + 0.05 \times 0.01} \approx 0.00058$$

Q: According to flightstats.com, American Airlines flights from Dallas to Chicago are on time 80% of the time. If 15 flights are randomly selected, what is the expected number of on-time flights?

$$E[X] = n p = 15 \times 0.8 = 12$$

Q: You survey 500 college students and find that 20 of them lie frequently. If the true proportion is 3%, what is the mean and standard deviation for the number of liars in samples of 500?

Mean: $$\mu = 500 \times 0.03 = 15$$, SD: $$\sigma = \sqrt{500 \times 0.03 \times 0.97} \approx 3.8$$

Question 1

A batch of fertilized chicken eggs has incubation times that are approximately normally distributed with a mean of 21 days and a standard deviation of 1 day. What is the z-score for an egg that hatches in 19 days?

Accepted Answer

$$z = \frac{19 - 21}{1} = -2$$

Question 2

Suppose the mean incubation time for fertilized chicken eggs is 21 days with a standard deviation of 1 day. What is the probability that a randomly selected egg hatches in less than 20 days? (Assume normality.)

Accepted Answer

About 15.87%

Question 3

For a normal distribution with mean $$\mu = 21$$ and $$\sigma = 1$$, what incubation time corresponds to the 17th percentile?

Accepted Answer

$$x = 21 + (z_{0.17})(1) \approx 21 + (-0.954) = 20.05$$

Question 4

A polygraph test for espionage returns a positive result 90% of the time if the person is guilty, and 1% of the time if the person is innocent. If 95% of suspects are actually guilty, what is the probability that a person is innocent given a positive polygraph result?

Accepted Answer

$$P(	ext{Innocent} | +) = \frac{0.05 	imes 0.01}{0.95 	imes 0.9 + 0.05 	imes 0.01} \approx 0.00058$$

Question 5

According to flightstats.com, American Airlines flights from Dallas to Chicago are on time 80% of the time. If 15 flights are randomly selected, what is the expected number of on-time flights?

Accepted Answer

$$E[X] = n p = 15 	imes 0.8 = 12$$

Question 6

In a binomial experiment with $$n = 15$$ and $$p = 0.8$$, what is the probability that exactly 10 flights are on time? (Use the binomial formula.)

Accepted Answer

$$P(X = 10) = \binom{15}{10} (0.8$$)^{10}$$ (0.2)^5$

Question 7

You survey 500 college students and find that 20 of them lie frequently. If the true proportion is 3%, what is the mean and standard deviation for the number of liars in samples of 500?

Accepted Answer

Mean: $$\mu = 500 	imes 0.03 = 15$$, SD: $$\sigma = \sqrt{500 	imes 0.03 	imes 0.97} \approx 3.8$$

Question 8

For the scenario above, use the normal approximation to estimate the probability that at least 20 out of 500 students lie frequently. (Use continuity correction.)

Accepted Answer

$$P(X \geq 20) \approx P\left(Z \geq \frac{19.5 - 15}{3.8}\right) \approx P(Z \geq 1.18) \approx 0.119$$

Question 9

Which of the following is NOT a requirement for using the normal approximation to the binomial distribution?

Accepted Answer

The probability $$p$$ must be close to 0.5

Question 10

If the middle 95% of incubation times for fertilized chicken eggs are desired, what z-scores correspond to the lower and upper bounds?

Accepted Answer

$$z = -1.96$$ and $$z = 1.96$$

Question 11

A binomial experiment is characterized by which of the following?

Accepted Answer

A fixed number of independent trials, each with two possible outcomes and constant probability of success

Question 12

Suppose the mean incubation time for eggs is 21 days and the standard deviation is 1 day. What is the probability that an egg hatches between 19 and 22 days?

Accepted Answer

$$P(19 \u003c X \u003c 22) = P(-2 \u003c Z \u003c 1) = \Phi(1) - \Phi(-2) \approx 0.8413 - 0.0228 = 0.8185$$

Question 13

If a randomly selected egg hatches in 18 days, would this be considered unusual? (Mean = 21, SD = 1)

Accepted Answer

Yes, because $$z = \frac{18 - 21}{1} = -3$$, which is more than 2 standard deviations from the mean

Question 14

In the context of Bayes’ Rule, which of the following best describes the 'posterior probability'?

Accepted Answer

The probability of a hypothesis given observed evidence

Question 15

A binomial random variable $$X$$ has $$n = 15$$ and $$p = 0.8. $$What is the variance of $$X$$?

Accepted Answer

$$Var(X) = n p (1-p) = 15 	imes 0.8 	imes 0.2 = 2.4$$

Question 16

If the probability that a flight is on time is 0.8, what is the probability that in 15 flights, at least 10 are on time?

Accepted Answer

$$P(X \geq 10) = \sum_{k=10}^{15} \binom{15}{k} (0.8)^k (0.2$$)^{15-k}$$

Question 17

Which of the following scenarios is best modeled by a binomial distribution?

Accepted Answer

Counting the number of heads in 20 flips of a fair coin

Question 18

If the mean incubation time is 21 days and the standard deviation is 1 day, what is the incubation time that separates the top 5% of eggs (i.e., the 95th percentile)?

Accepted Answer

$$x = 21 + (1.645)(1) = 22.645$$

Statistics Review: Probability, Binomial Distributions, Normal Distributions, and Bayes' Rule

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