According to U.S. News & World Report's publication America's Best Colleges, the average cost to attend the University of Southern California (USC) after deducting grants based on need is $26,950. Assume the population standard deviation is $7,800. Suppose that a random sample of 70 USC students will be taken from this population. Use z-table.

a. What is the value of the standard error of the mean?

(to nearest whole number)

b. What is the probability that the sample mean will be more than $26,950?

(to 2 decimals)

c. What is the probability that the sample mean will be within $750 of the population mean?

(to 4 decimals)

d. How would the probability in part (c) change if the sample size were increased to 160 ?

(to 4 decimals)

A survey on British Social Attitudes asked respondents if they had ever boycotted goods for ethical reasons (Statesman, January 28, 2008). The survey found that 23% of the respondents have boycotted goods for ethical reasons.

1. In a sample of six British citizens, what is the probability that two have ever boycotted goods for ethical reasons?
2. In a sample of six British citizens, what is the probability that at least two respondents have boycotted goods for ethical reasons?
3. In a sample of ten British citizens, what is the probability that between 3 and 6 have boycotted goods for ethical reasons?
4. In a sample of ten British citizens, what is the expected number of people that have boycotted goods for ethical reasons? Also find the standard deviation.

1. Let’s assume you have a bag of balls (5 red and 5 black). You roll a six-sided dice and then select a ball from the bag.
   1. What is the probability you roll a five on the die and then select a red ball?
2. You are playing blackjack with no other players (i.e. cards are dealt immediately with no one else receiving cards). What is the probability you get a K, Q, J, or 10 on the first card followed by an ace on the second card? (No replacement)
3. The Powerball Lottery requires you to select 5 numbers between 1 and 69 and 1 number between 1 and 26. What is the probability of selecting ALL numbers correct? (Please show your work).

The table below shows the number of deaths in the U.S. in a year due to a variety of causes. For these questions, assume these values are not changing from year to year, and that the population of the United States is 312 million people.

Make sure your answer is accurate to at least 2 significant figures (values after leading zeros)

c) What is the probability that you will die as a passenger car occupant next year?

c) What is the probability that you will died as a passenger car occupant last year?

a) What is the probability that an American chosen at random died as a passenger car occupant last year?

1. Based on historical data, your manager believes that 31% of the company's orders come from first-time customers. A random sample of 82 orders will be used to estimate the proportion of first-time-customers. What is the probability that the sample proportion is less than 0.28?

Answer =  (Enter your answer as a number accurate to 4 decimal places.)

2. In a recent year, the Better Business Bureau settled 75% of complaints they received. (Source: USA Today, March 2, 2009) You have been hired by the Bureau to investigate complaints this year involving computer stores. You plan to select a random sample of complaints to estimate the proportion of complaints the Bureau is able to settle. Assume the population proportion of complaints settled for the computer stores is the 0.75, as mentioned above. Suppose your sample size is 102. What is the probability that the sample proportion will be at least 3 percent more than the population proportion?

Answer =  (Enter your answer as a number accurate to 4 decimal places.)

3. Based on historical data, your manager believes that 27% of the company's orders come from first-time customers. A random sample of 186 orders will be used to estimate the proportion of first-time-customers. What is the probability that the sample proportion is less than 0.32?


Answer =  (Enter your answer as a number accurate to 4 decimal places.)

To test whether my friend’s fish Googly had psychic powers, I wrote R code to display two windows. I entered either “Left” or “Right” depending on which way Googly was facing. Then the random number generator in R selected either the left or the right window, with probability 0.5 for each, in which to display a star. Let p be the probability Googly guesses correctly on a given trial (assume this is constant.) In 80 trials, Googly correctly guessed the window with the star 41 times. (a) (3 points) Using mathematical notation, write down null and alternative hypotheses for a one-sided test. (b) (3 points) If the test statistic is the number of correct guesses (41) in 80 trials, write down R code to find the P-value of a one-sided test. (c) (2 points) Even without R, we can see that Googly’s success rate was close to its expected value under the null, so the one-tailed P-value will be close to 0.5. State your conclusion about the fish’s psychic powers. (d) (2 points) Continued from part (b). If you only known that the R code dbinom(40, 80, 0.5) gives the number 0.0889, how would you find the exact P-value of the test? (Hint: use the property of a symmetric probability distribution.)

Partial Permutations

Find the number of 7-character (capital letter or digit) license plates possible if no character can repeat and: a) there are no further restrictions, b) the first 3 characters are letters and the last 4 are numbers, c) letters and numbers alternate, for example A3B9D7Q or 0Z3Q4A9

Combinations

A standard 52-card deck consists of 4 suits and 13 ranks. Find the number of 5-card hands where: a) any hand is allowed (namely the number of different hands) b) all five cards are of same suit c) all four suits are present d) all cards are of distinct ranks

Distribution Types

1) Which of the following sample spaces are uniform?

a) {land,sea} for a randomly point on a globe b) {odd, even} for a random integer from {1,2,. . . ,100} c) {leap year, non-leap year} for a random year before 2019 d) {two he{distance to origin} for a random point in {−3, −1, 1, 3} × {−4, −2, 2, 4} e) lads, two tails, one head and one tail} when flipping two fair coin

Inequalities

.in any uniform probability space: a) ?⊇? ⟶ ?(?)≥?(?) b) ?(?)≥?(?) ⟶ ?⊇? c) |?|≥|?| ⟶ ?(?)≥?(?) d) ?(?)≥?(?) ⟶ |?|≥|?|

Conditional Probability

Three fair coins are sequentially tossed. Find the probability that all are heads if: a) the first is tails b) the first is heads c) at least one is heads.

a. Consider a regular 52-card deck. If you pick a card randomly and then replace it 5 times, what is the probability of getting an Ace exactly two times?

b. Consider flipping a fair coin 10 times. Let X(t) be the number of heads when the outcome is t. What is P(X = 0 or X = 7)?

c.Consider a coin that is weighted so that that flipping it results in heads 3 times as often as tails. If you flip the coin 10 times, what is the probability of getting exactly 5 heads given that the first two flips come up heads.
d. Consider two dice. Die 1 is a regular, fair, 6-sided die. Die 2 is weighted so that a 6 gets rolled 3 times as often as any other number. If you roll each die three times, how many times more likely are you to get 3 6's with the weighted die than with the fair die? [That is, what is P(three 6's with Die 2)/P(three 6's with Die 1)?] Round your answer to the nearest whole number.
e. if you flip a fair coin 10 times, what is the probability of getting at least 5 heads given that the first 3 flips come up heads