The longest "run" of S's in the sequence SSFSSSSFFS has length 4, corresponding to the S's on the fourth, fifth, sixth, and seventh positions. Consider a binomial experiment with n = 4, and let y be the length (number of trials) in the longest run of S's. (Round your answers to four decimal places.)

(a) When p = 0.5, the 16 possible outcomes are equally likely. Determine the probability distribution of y in this case (first list all outcomes and the y value for each one).

Calculate μ_y.

(b) Repeat Part (a) for the case p = 0.4.

Calculate μ_y.

According to a recent survey conducted at a local college, we found students spend an average of 19.5 hours on their smartphone per week, with a standard deviation of 3.5 hours. Assuming the data follows the normal distribution.

a) How many percent of students in this college spend more than 15 hours on their smartphone per week?

b) If we randomly select 12 students, what is the probability that the average of these students spending on the internet is more than 20 hours per week?

c) What is the 36th percentile for number of hours spending on their smartphone per week?

A pizza delivery company classifies its customers by gender and location of residence. The research department has gathered data from a random sample of 2001 customers. The data is summarized in the table below.

Gender and Residence of Customers

Residence Males Females

Apartment 233 208

Dorm 159 138

With Parent(s) 102 280

Sorority/Fraternity House 220 265

Other 250 146

What is the probability that a customer is male and lives in 'Other' or is male and lives in a fraternity house? Express your answer as a fraction or a decimal number rounded to four decimal places.

We are testing a new prescription drug that cures the common cold. One of the side effects of the drug is insomnia (can’t sleep). There is a 6% chance the patient will develop insomnia. We have selected 50 people and administered the drug to them and asked if they developed insomnia. Define the random variable x to be the number of people who develop insomnia.

Showing your work in Excel as demonstrated in class, what is the probability that x is:

Exactly 3?

Exactly 6?

Greater than or equal to 5?

Greater than 5?

Less than or equal to 7?

Less than 7?

A researcher hypothesizes that electrical stimulation of a region of the ventromedial hypothalamus will result in appetite suppression. To test this, one set of rats is given electrical stimulation, while a control group does not. Rats are subsequently given unrestricted access to sugar pellets for 5 minutes, and the number of pellets eaten is recorded.

a. Use the appropriate test of hypothesis to determine whether or not the treatment rats consume fewer pellets. Use α = .05

No Stimulation: 12, 7, 3, 11, 8,5,14,7,9,10
Stimulation: 8,7,4,14,6,7,12,5,5

b. Based on these findings, what is the exact probability of a Type 1 error occurring?

Q2.Calculate the distance to default if the current market value of asset is 300 Million, growth is 3%, asset volatility is 15%, current liabilities are 200 Million, and non-current liabilities are 140 Million, interpret your answer.

If there are 500 firms with this distance to default, and 30 of these have defaulted, calculate the expected default probability, explain in detail the implications this number has for the future default rate?

Distance to default = ((Market Value of Assets) – (Current Liabilities + 0.5 X Non Current Liabilities))/(Market Value of Assets X Asset Volatility)

1. Do each of the following:

a) What is the minimum number of people that must be in a room to guarantee that at least 20 were born on the same day of the week? Assume all days of the week are equally likely.

b) What is the coefficient of x 9 y 11in ( 3 x − 4 y ) 20?

c) What is the probability that in a permutation of the letters {A, B, C, D, E, F, G, H, I, J, K, L}, the strings BAIL and ILK appear?

d) How many decimal strings of length 15 begin with 321 or end with 9768?

1) Why would one use the median instead of the mean?

2) An experiment involves rolling a six-sided die 480 times and recording the number of 3s. What is the mean and standard deviation?

3) In a normal distribution, what is the percentage of data having a z score less than -1?

4) We roll 5 six-sided die. What is the probability of obtaining exactly two 1s?

5) What is sampling error, and how would you distinguish it from nonsampling error?

6) What types of inferences can we make?

Questions

1. [10 marks] For each of the following statements, identify whether the statement is true or false, and explain why. Please limit each response to no more than 2 sentences.

a) The probability that a continuous random variable takes a specific value is 0.

b) Statistical inference is the process of drawing conclusions about unknown statistics by using known parameters.

c) The terms “histogram” and “bar graph” are synonyms.

d) If two events A and B are independent, then there is no overlap between the events on a Venn diagram.

e) The standard deviation of the number 10 is 1.

Let X equal the number of chocalate chips in a chocalate chip cookie. Sixty-two oberservations of X yielded the following frequencies for the possible outcomes of X:

(a) Use this data to graph the relative frequency histogram and the Poisson probability with

lamda = 5.6 on the same figure.

(b) Do these data seem to be observations of a Poisson random variable with

lamda = 5.6 ?