Identify the distribution and give a symbolic expression for each indicated probability, identifying parameters. Then use Mathematica or Excel to evaluate the indicated probabilities.

1. In a large biochemistry class of 41 chemistry and 81 biology majors, 7 students are selected at random to prepare a presentation. What is the probability that
   1. 2 chem and 5 bio students are selected? (b) At least one chemistry major is selected?
2. Blood tests of winning horses in thoroughbred races disqualify 9% of them for use of illegal medications. What is the likelihood that
   1. No horse is disqualified in this weekend’s 19 races?
   2. The first disqualified winning horse occurs between the 23^rd and 31^st race (inclusive)?
3. During rush hour, cars pass a given point on the parkway at the instantaneous rate of 98 per minute.
   1. What is the probability that at least 430 pass that point in a five-minute period?
4. In a state that requires licenses for Physician Assistants, 39% of applicants pass the licensing exam on the first try. If 2680 students took the exam for the first time the last time it was administered,
   1. What is the likelihood that at most 1000 passed?
5. 6.25% of logins to the Seton Hall website fail. Assuming attempts are independent:
   1. What is the probability that the 10^th failure occurs on the 140^th login?
   2. What is the probability that the 6^th failure has not occurred in the first 100 logins?
6. Individuals who barely survive major disasters, on average, suffer nightmares during sleep every 2.75 hours [during the first month].
   1. Find the probability that someone’s 10^th nightmare occurs within the first 40 hours of sleep.

1. For each of the derivations above, give the mean and standard deviation. Use the formulas from the notes. For Geometric and Negative Binomial, be careful about variations.

You own a small company. Last year you conducted a study to learn more about your customers. You found that the mean age of your customers was 31.84 years with a standard deviation of 9.84 years. This year you take a random sample of 60 customers. What is the probability that the mean age of those 60 customers is greater than 33 years?

Three students accidentally leave copies of the textbook in their classroom after class. At the beginning of the next lecture, the professor distributes the three books in a completely random fashion to each of the three students (1, 2, and 3) who left their books. One possible outcome is that 1 receives 2’s book, 2 receives 1’s book, 3 receives his or her own book. This outcome can be abbreviated as (2, 1, 3).

a. List all other possible outcomes.

b. Let Y denote the number of students who receive their own book. Determine the PMF of Y.

A manufacturer of a specific pesticide useful in control of household insects claims that after six months on the shelf, the variance of the amount of active ingredient among cans is no more than 4 grams2. A consumer group obtained a random sample of 20 recently produced cans of the pesticide from the manufacturer. The cans were stored for 6 months and then individually tested for the amount of active ingredient. The sample variance of the 20 cans was 6.2 grams2. a. Is there sufficient evidence to indicate that the population variance of active ingredient has more variability after 6 months than that claimed by the manufacturer? Use α=.05. b. To what population does your conclusion formally apply? What distributional assumptions must be made about the amount of active ingredient in a can of pesticide?

What Influences the Sample Size? We examine the effect of different inputs on determining the sample size needed to obtain a specific margin of error when finding a confidence interval for a proportion. Find the sample size needed to give, with 95% confidence, a margin of error within plus-or-minus 2% when estimating a proportion. First, find the sample size needed if we have no prior knowledge about the population proportion p. Then find the sample size needed if we have reason to believe that p almost-equals 0.7. Finally, find the sample size needed if we assume p almost-equals 0.8. Round your answers up to the nearest integer.

Population proportion Sample Size

No knowledge:

0.7:

0.8:

The mean number of sick days an employee takes per year is believed to be about 10. Members of a personnel department do not believe this figure. They randomly survey 8 employees. The number of sick days they took for the past year are as follows: 12; 6; 14; 3; 11; 9; 7; 9. Let X = the number of sick days they took for the past year. Should the personnel team believe that the mean number is about 10? Conduct a hypothesis test at the 5% level. State the distribution to use for the test. What is the test statistic? What is the p-value? Construct a 95% confidence interval for the true mean. Sketch the graph of the situation. Label the point estimate and the lower and upper bounds of the confidence interval. (Round your answers to three decimal places.) **Please use a TI*$ Plus where possible**

Professor Fair believes extra time does not improve grades on exams. He randomly divided a group of 300 students into two groups and gave them all the same test. One group had exactly 1 hour in which to finish the test, and the other group could stay as long as desired. The results follow. Test at the 0.01 level of significance that time to do a test and test results are independent.

A. What is the null hypothesis?

B. What is the alternative hypothesis?

C. What distribution are you using?

D. What test are you running?

E. What is your conclusion?

Please answer all parts of the question, with all work shown

Suppose there are 300 cards in a box numbered 1 through 300. Therefore, the number of each card has one, two, or three digits. A card is drawn at random from the box. Suppose that the number on the card has X digits of which Y are 0. Suppose we would like to know whether X and Y are correlated, negatively correlated, or uncorrelated. Determine Cov(X,Y)Cov(X,Y) and ρ(X,Y)ρ(X,Y), hence settling the question.

"Unknown cultural affiliations and loss of identity at high elevations." These are words used to propose the hypothesis that archaeological sites tend to lose their identity as altitude extremes are reached. This idea is based on the notion that prehistoric people tended not to take trade wares to temporary settings and/or isolated areas. As elevation zones of prehistoric people (in what is now the state of New Mexico) increased, there seemed to be a loss of artifact identification. Consider the following information. Elevation Zone Number of Artifacts Number Unidentified 7000-7500 ft 113 73 5000-5500 ft 145 20 Let p1 be the population proportion of unidentified archaeological artifacts at the elevation zone 7000-7500 feet in the given archaeological area. Let p2 be the population proportion of unidentified archaeological artifacts at the elevation zone 5000-5500 feet in the given archaeological area. (a) Find a 95% confidence interval for p1 − p2. (Round your answers to three decimal places.)

lower limit=

upper limit=

Suppose individual X scores in the population follow a normal distribution N(38, 20). A researcher draws numerous samples of sample size n = 100 from the population, and in each sample, she calculates the sample mean. Then 68% of these sample means should approximately fall within Group of answer choices (A) 34 and 40 (B) 34 and 38 (C) 38 and 44 (D) 36 and 40

In this hypothetical case study, a new rapid test kit nicknamed Alpha is being reviewed. The sensitivity of Alpha is 93.0% (0.93) and the specificity is 96.0% (0.96). Assume that the actual prevalence of the Zika antibody among the United States population of blood donors is 4% (0.04) and that of the hurricane disaster relief volunteers returning from Puerto Rico is 20.0% (0.20).

Construct a separate 2 x 2 table, to calculate the PPV and NPV for a population of 2,500 volunteers who aided in the recent hurricane relief efforts in the Caribbean.

If sensitivity and specificity remain constant, what is the relationship of prevalence to predictive-value positive and predictive-value negative? (Hint: Think if one increases, decreases, or stays the same.)

Let X~Bin(100,0.5).

Show all workings in details

a) Find the probability that X is a perfect square.

b) Find the probability that X is a greater than 60.

c) Find the expected value and variance of X.

1. A pharmacy is using X bar and R charts to record the time it takes to fill a prescription after the customer has turned in or called in the prescription. Each day, the pharmacy records the times it takes to fill six prescriptions. During a 30-day period, the hospital obtained the following values: X double bar = 20 minutes; R bar = 4 minutes. The upper and lower specifications are 23 minutes and 13 minutes respectively. What is the value of 6 sigma, to one decimal place?

Based on a recent​ study, the pH level of the arterial cord​ (one vessel in the umbilical​ cord) is normally distributed with mean 7.36 and standard deviation of 0.11. Find the percentage of preterm infants who have the following arterial cord pH levels.

a. pH levels between 7.00 and 7.50.

b. pH levels over 7.44.

A pharmaceutical company is testing a new drug to increase memorization ability. It takes a sample of individuals and splits them randomly into two groups. After the drug regimen is completed, all members of the study are given a test for memorization ability with higher scores representing a better ability to memorize. Those 23 participants on the drug had an average test score of 31.622 (SD = 4.794) while those 21 participants not on the drug had an average score of 32.04 (SD = 5.335). You use this information to create a 95% confidence interval for the difference in average test score. What is the margin of error? Assume the population standard deviations are equal.