AP Stats Probability #32-50

You win a game if you flip a coin with heads coming up exactly 50 percent of the tosses. Would you rather flip 10 times or 100 times? (A) 10 times because .246>.080 (B) 10 times because of the Central Limit Theorem (C) 100 times because ₁₀₀c₅₀>₁₀c₅ (D) 100 times because of the Law of Large Numbers (E) Your chance of winning is the same with 10 or 100 flips

(A) ₁₀c₅(.5)⁵(.5)⁵= .246 while ₁₀₀c₅₀(.5)⁵⁰(.5)⁵⁰=.080. Or on the TI-84, binompdf (10, .5, 5) =.246 while binompdf (100, .5, 50)= .080. The Law of Large Numbers talks about relative frequency in the long run becoming close to an expected probability, but not about getting that exact probability.

Given the probabilities P(A) = .3 and P(A ∪ B) = .7, what is the probability P(B) if A and B are mutually exclusive? If A and B are independent? (A) .4, .3 (B) .4, 4/7 (C) 4/7, .4 (D) .7, 4/7 (E) .7, .3

(B) If A and B are mutually exclusive, P(A ⌒ B) = 0. Thus, .7 = .3 + P(B) - 0, and so p(B) = .4. If A and B are independent, then P(A ⌒ B) + P(A)P(B). Thus .7 = .3 + P(B), and so P(B) = 4/7.

It is estimated that 20 percent of all drivers do not signal when changing lanes. In a random sample of four drivers, what is the probability that at least one doesn't signal when changing lanes? (A) 1 - (.2)⁴ (B) 1 - (.8)⁴ (C) 4(.2)(.8)³ (D) 4(.2)³(.8) (E) 4(.2)³(.8) + 6(.2)²(.8)² + 4(.2)(.8)³ + (.8)⁴

(B) In this binomial situation, the probability that a driver does signal when changing lanes is 1 - .2 = .8, the probability that all of the drivers signal when changing lanes is (.8)⁴, and thus the probability that at least one doesn't signal when changing lanes is 1 - (.8)⁴.

A basketball player makes one out of his first two free throws. From that point on, the probability that he makes the next shot is equal to the proportion of shots made up to that point. If he takes two more shots, what is the probability he ends up making a total of two free throws? (A) 1/4 (B) 1/3 (C) 1/2 (D) 2/3 (E) 3/4

(B) Needs picture

Which of the following statement is a true statement? (A) The area under the standard normal curve between 0 and 2 is twice the area between 0 and 1 (B) The area under the standard normal curve between 0 and 2 is half the area between -2 and 2 (C) For the standard normal curve, the interquartile range is approximately 3. (D) For the standard normal curve, approximately 1 out of 1000 values are greater than 10 (E) The 68-95-99.7 rule applies only to normal curves where the mean and standard deviation are known.

(B) Statement (B) is true by symmetry, statement (A) is false because .4772 is not 2 times .3413 and statement (C) is false because 2 times 0.67 is not 3. Less than 10⁻²⁰ ( a very small number!) values are greater than a standardized z-score of 10. The 68-95-99.7 rule (the Empirical Rule) applies to all normal distributions.

Supposed we have a random variable X where for the calues k = 0, ... , 12, the associated probabilities are (12 on top of k)(.34)^k(.66)¹²-k. What is the mean of X> (A) 0.34 (B) 0.66 (C) 4.08 (D) 7.92 (E) None of the above

(C) This is a binomial with n = 12 and p = .34, and so the mean is np = (12)(.34) = 4.08

A population is normally distributed with mean 58. Consider all samples of size 5. The variable x-bar - 58/ s/√5 (A) has a normal distribution (B) has a t-distribution with df = 5 (C) has a t-distribution with df = 4 (D) has neither a normal distribution nor a t-distribution. (E) has either a normal distribution or a t-distribution depending on the characteristics of the population standard deviation.

(C) x-bar - mu/ s/√n has a t-distribution with df = n - 1

Which of the following statements is incorrect? (A) Sample statistics are used to make inferences about population parameters. (B) Statistics from smaller samples have more variability. (C) Parameters are fixed, while statistics vary depending on which sample is chosen (D) As the sample size n becomes larger, the sample distribution becomes closer to a normal distribution. (E) All of the above are true statements

(D) As the sample size n becomes larger, the sample distribution becomes closer to the population distribution ( the sampling distribution may become closer to a normal distribution)

Suppose 56 percent of eight to twelve year olds expect to have a "Great Life". In an SRS of 125 eight eight to twelve year olds, what is the probability that between 50 percent and 60 percent will say they expect to have "a great life". (A) .2721 (B) .5402 (C) .6723 (D) .7279 (E) .8640

(D) Both np= 125(.56) = 70 > 10 and n(1 - p) = 125(.44) = 55 > 10. The sampling distribution of P∧ is approximately normal with mean .56 and standard deviation σp∧ = √(.56)(.44)/ 125 = .0444. The probability a sample proportion is between .50 and .60 is normalcdf (.5, .6, .56, .0444) = .7279.

The waiting time for a commuter bus is normally distributed with a mean of 8 minutes and a standard deviation of 2 minutes. If there are 3,000 riders a day, which of the following is the shortest time interval associated with 2,000 riders? (A) 0 to 8.9 minutes (B) 6.1 to 7.1 minutes (C) 6.1 to 8 minutes (D) 6.1 to 9.9 minutes (E) 7.1 to 14 minutes

(D) From the shape of the normal curve, the answer is in the middle. The middle two-thirds, leaving one-sixth in each tail, is between z-scores of ±invNorm (5/6) = ± 0.9674, and 8 ± 2(0.9674) gives (6.1, 9.9)

The average noise level in a bar is 36 decibels with a standard deviation of 5 decibels. Assuming a normal distribution, what is the probability the noise level is between 30 and 40 decibels? (A) .327 (B) .337 (C) .381 (D) .673 (E) .683

(D) On the TI-84, normalcdf(30, 40, 36, 5) = .673

A company has a choice of three investments schemes. Option 1 gives a sure $30,000 return on investment. Option 2 gives a 50 percent chance of returning $50,000 and a 50 percent chance of returning nothing. Option 3 gives a 10 percent chance of returning $100,000 and a 90 percent chance of returning nothing. Which option should the company choose? (A) Option 1 if it wants to maximize expected return (B) Option 2 if it needs at least $40,000 to pay off an overdue loan (C) Option 3 if it needs at least $75,000 to pay off an overdue loan (D) All of the above answers are correct (E) Because of chance, it really doesn't matter which option it chooses.

(D) Option 1 gives the highest expected return: 30,000 is greater than both 50,000(.5) = 25,000 and 100,000(.10) = 10,000. Option 2 gives the best chance (.50) of paying off the $40,000 loan, Option 3 gives the only chance of paying off the $75,000 loan. The moral is that the greatest expected value is not automatically the "best" answer.

The Air Force receives 40 percent of it parachutes from company C₁ and the rest from company C₂. The probability that a parachute will fail to open is .0025 or .002, depending on whether it is from company C₁ or C₂, respectively. If a random chosen parachute fails to open, what is the probability that it is from company C₁? (A) .0010 (B) .0022 (C) .4025 (D) .4545 (E) .5455

(D) P(C₂) = 1 - P(C₁) = 1 = .40 = .60 Picture needed here P(Fails) = .001 + .0012 = .0022. P(C₁ | fails) = P(C₁ ⌒ fails) / P(fails) = .001/.0022 = .4545

Which of the following is not a valid discrete probability distribution for the set (x₁, x₂. x₃)? (A) P(x₁)= 1, P(x₂)=0, p(x₃)=0 (B) P(x₁)=1/3, P(x₂)=1/3, P(x₃)=1/3 (C) P(X₁)=½, P(X₂)=1/3, P(X₃)= 1/6 (D) P(X₁)= 2/3, P(X₂)=2/3, P(X₃)= -1/3 (E) All the above are valid probability distributions

(D) The Probabilities must sum to 1, and no individual probability can be negative

There are 8,253 men and 10,327 women at a state university. If 43 percent of the men and 27 percent of the women are business majors, what is the expected number of business majors in a random sample of 200 students? (A) 31,7 (B) 34.1 (C) 63.4 (D) 68.2 (E) 70.0

(D) The probability a student is male is 8,253/18,580 = .4442 and female is 1−.4442 = .5558. We thus calculate 200(.4442)(.43) + 200(.5558)(.27) = 68.2

Which of the following statements is a true statement? (A) The sampling distribution of p-hat has a mean that can vary from the population proportion p by approximately 1.96 standard deviations. (B) The sampling distributions of p-hat has a standard deviation equal to √np(1 - p) (C) The sampling distribution of p-hat is considered close to normal provided that n ≥ 30. (D) The sample proportion is a random variable with a probability distribution. (E) All of the above are true statements

(D) The sample proportion is a random variable with a probability distribution called the sampling distribution of p-hat. The sampling distribution of p-hat has a mean equal to the population proportion p, a standard deviation equal to √p(1 - p)/ n , and is considered close to normal provided that both np and n(1 - p) are large enough ( greater than 5 or 10 are standard guidelines). The sampling distribution of x-bar is usually close to normal when n≥ 30.

Given a random variable X taking three possible values x₁, x₂, x₃, which of the following statements must be true? (A) x₁ + x₂ + x₃ = 1 (B) E(X) = 1/3∑xi (C) var(X) = 1/3∑(xi - x-bar)² (D) E(X + c) = E(X) + c (E) var(aX) + a var(X)

(D) The sum of the probabilities of the xi equals 1, but the sum of the xi themselves can equal anything. E(X) = ∑ xi P(xi) and var(X) = ∑(xi - x-bar)² P(xi). Furthermore, X(X + c) = E(X) + c, but var(aX) = a² var(X)

Which of the following statements is incorrect? (A) Like the normal, t-distributions are always unimodal (B) Like the normal, t-distributions are always symmetric. (C) Like the normal, t-distributions are always bell shaped (D) The t-distributions have less spread than the normal; that is, they have less probability in the tails and more in the center than the normal. (E) For larger values of df, degrees of freedom, the t-distributions look more like the normal distributions

(D) The t-distributions are unimodal, symmetric, bell-shaped, and approach the normal for large df, but they have more, not less, spread than the normal distribution.

Which of the following are unbiased estimators for the corresponding population parameters? I. Sample means] II. Sample proportions III. Difference of sample means IV. Difference of sample proportions (A) None are unbiased (B) I and II (C) I and III (D) III and IV (E) All are unbiased

(E) All are unbiased estimators for the corresponding population parameters, that is, the means of their sampling distributions are equal to the population parameters

Populations P₁ and P₂ are normally distributed and have identical means. However, the standard deviation of P₁ is twice the standard deviation of P₂. What can be said about the percentage of observations falling within two standard deviations of the mean for each population? (A) The percentage for P₁ is twice the percentage for P₂ (B) The percentage for P₁ is greater, but not twice as great, as the percentage for P₂ (C) The percentage for P₂ is twice the percentage for P₁ (D) The percentage for P₂ is greater, but not twice as great, as the percentage for P₁ (E) The percentages are identical

(E) All normal distributions have about 95 percent of their observations within two standard deviations of the mean.

(A) When events are independent, the probability of their intersection is the product of their probabilities. In this case, (3/4)(.10)(.35) = .02625

Three-fourths of college students change their major at least once. The reasons for changing and what they change to are as follows: Assuming reasons and what students change to are independent, what is the probability that a college student decides to change a major, bases on advisor suggestion, to pre-professional? (A) .0263 (B) .0350 (C) .0467 (D) .3000 (E) .4500