There is a special calculator that when used to add numbers up, it always rounds up or down the number to the closest integer and ignore the decimals. For example, you want to add 1.1+2+3.6, it will show can calculate as 1+2+4, etc. Assume the error from each adding follows uniform distribution on (-0.5, 0.5).

1. When adding 1500 numbers together, what is the probability of the absolute value of total error is greater than 15?
2. How many numbers can you add until the probability of the absolute value of total error being less than 10 reaches 0.9?

(hint, you may need to find the expectation and the variance of uniform distribution, and using CLT for your calculation)

A philosophy professor decides to give a 20 question multiple-choice quiz to determine who has read an assignment. Each question has 3 choices. Let Y be the random variable that counts the number of questions that a student guesses correctly. You can assume that questions and answers are independent.

1. Find the probability distribution function of Y by making a table of the possible values of Y and their corresponding probabilities.

2. Add a column for the Cumulative Distribution Function (CDF) for each value of X.

3. Find the expected value and standard deviation of Y.

4. If she wants to choose a passing grade such that the probability of passing a student who guesses at every question is less than 0.05, then what score should she set as the lowest passing grade?

A leading magazine (like Barron's) reported at one time that the average number of weeks an individual is unemployed is 40 weeks. Assume that for the population of all unemployed individuals the population mean length of unemployment is 40 weeks and that the population standard deviation is 2.8 weeks. Suppose you would like to select a random sample of 92 unemployed individuals for a follow-up study.

Find the probability that a single randomly selected value is greater than 40.6. P(X > 40.6) = (Enter your answers as numbers accurate to 4 decimal places.)

Find the probability that a sample of size n = 92 is randomly selected with a mean greater than 40.6. P(M > 40.6) = (Enter your answers as numbers accurate to 4 decimal places.)

1- The mean incubation time of fertilized eggs is 19 days. Suppose the incubation times are approximately normally distributed with a standard deviation of 1 day.

​Q: The incubation times that make up the middle 39​% are ____ to ____ days.

​(Round to the nearest whole number as needed. Use ascending​ order.)

2- A study found that the mean amount of time cars spent in​ drive-throughs of a certain​ fast-food restaurant was

141.7 seconds. Assuming​ drive-through times are normally distributed with a standard deviation of 27 ​seconds.

Q: The probability that a car spends more than 3 minutes in the​ restaurant's drive-through is ____​, so it(would, would not be)​ unusual since the probability is(less, greater) than 0.05.

​(Round to four decimal places as​ needed.)

Which of the following statements concerning financial risk is false?

              a.           Generically, financial risk is related to the probability of a return less than expected.

              b.           If an investment is held in isolation (stand-alone), the appropriate measure of risk is the beta coefficient.

              c.           In theory, you can create a riskless portfolio by combining a large number of investments whose returns are uncorrelated (independent of one another).

              d.           In the real world, it is not possible to create a riskless portfolio because all investment returns, to a greater or lesser extent, move with the overall economy.

              e.           Assume you know for certain that an investment will return negative 10 percent. (In other words, the probability of a -10% return is 100 percent.) Although the expected return is negative, the investment is riskless.

Suppose the number of French fries in a small bag at a particular fast-food restaurant follows a normal model with mean 49 fries and standard deviation 5 fries.

1. Estimate the proportion of the bags that contain at least 40 fries.

2. Estimate the proportion of the bags that contain 50 or fewer fries.

3. The restaurant could advertise that 99% of its bags contain at least how many fries?

4. If the restaurant wants to increase the probability in Question 1 by changing the mean, should it increase or decrease the mean? Explain intuitively and draw a sketch.

5. If the restaurant wants to increase the probability in Question 1 by changing the standard deviation, should it increase or decrease the standard deviation? Explain intuitively, including a sketch but no calculations.

A computerized driving test held ten sessions in a day. The overall passing rate of the driving test is 87.5%. In each session, there are 14 candidates. (a) Most likely, how many candidates would pass the test in a session? What is the probability of its happening? (b) What is the probability that more than 10 candidates in a session can pass the test? (c) What are the (i) expectation, (ii) variance, and (iii) standard deviation of number of candidates can pass the driving test in a day. (d) Suppose each candidate who fail the driving test will immediately join the tutorial class for the 2nd test. The charge of the tutorial class is $400. What are the (i) expectation and (ii) standard deviation of the daily income earned from the tutorial class?

As the assistant bank manager, you want to provide prompt service for customers at your bank's drive thru window. The bank can currently serve up to 10 customers per 15-minute period without significant delay. The average arrival rate is 7 customers per 15-minute period. Let x denote the number of customers arriving per 15-minute period. Assume this is a Poisson distribution.

a. Find the probability that 10 customers will arrive in a particular 15-minute period.
b. Find the probability that 10 or fewer customers will arrive in a particular 15-minute period.
c. Does your own personal bank have prompt service? Are there ways that your bank can improve their customer service? How?

BINOMIAL PROBABILITIES Big Box Store (BBS) has an annual rate of 4% of all sales being returned. In a recent sample of thirty randomly selected sales the number of returns was five. BBS is concerned about the event, and your advice is solicited. ( FOR 4 – 9, OPEN THE EXCEL EXAM TWO FILE. THE SPREADSHEET FOR THIS PROBLEM IS FOUND ON SHEET TWO. COMPLETE AND SAVE YOUR WORK IN EXCEL AND ENTER THE SOLUTIONS BELOW. (6) (2.5points) What is the probability that a random sample of 30 sales has 5 or fewer returns? Type the answer and any work below.   (7) (2.5 points) What is the probability that a random sample of 30 sales has less than four returns? Type the answer and any work below.

Advertisers contract with Internet service providers and search engines to place ads on websites. They pay a fee based on the number of potential customers who click on their ad. Unfortunately, click fraud—the practice of someone clicking on an ad solely for the purpose of driving up advertising revenue—has become a problem. According to BusinessWeek 43% of advertisers claim they have been a victim of click fraud. Suppose a simple random sample of 300 advertisers will be taken to learn more about how they are affected by this practice. Use z-table.

a. What is the probability that the sample proportion will be within +- 0.03 of the population proportion experiencing click fraud?

(to 4 decimals)

b. What is the probability that the sample proportion will be greater than 0.49?

(to 4 decimals)