Please answer the attached basic stats questions and use R programming for the questions 5 and 6

Question 1. [20 points] The distribution of automobile speeds on a certain highway is normally distributed

with a mean of 71.6 miles/hour, and a standard deviation of 4.53 miles/hour.

(a) What is the probability that an automobile travels slower than 75 miles/hour?

(b) What is the probability that an automobile travels faster than 65 miles/hour?

(c) What is the probability that an automobile travels between 65 and 75 miles/hour?

(d) What is the 90th percentile for the speeds of automobiles on this highway?

2

Question 2. [10 points] Let X1, X2, · · · , X100 be a random sample of size n = 100 from a population with

mean µ = 20 and σ = 15. Let X¯ =

1

100

P100

i=1 Xi

.

(a) What is E(X¯) and SD(X¯) (i.e., the mean and standard deviation of the sampling distribution for X¯)?

(b) What distribution does the sample mean X¯ follow? Explain.

(c) Calculate P(X >¯ 23)

Question 3. [10 points] From past experience, it is known that the number of tickets purchased by each

customer standing in line at the ticket window at a certain movie theater follows a distribution with mean

µ = 1.5 and standard deviation σ = 0.6. Suppose that there are 40 customers waiting in line to get tickets

before the start of a movie. If only 55 tickets remain, what is the probability that all 40 customers will be

able to purchase the tickets they desire?

Question 4.[10 points] Mark the following as either True or False.

(a) The mean and median are usually approximately equal when the distribution of data in a

histogram is skewed right.

(b) The daily commute for an employee takes an average of 1.3 hours with a standard deviation of

0.45 hours. When converting from hours to minutes, the average commute time is 78 minutes with a

standard deviation of 27 minutes.

(c) The following scenario is an example of stratified random sampling: A polling agency is interested in determining how concerned adult Americans are about the coronavirus’s effect on the economy.

The agency decides to interview a simple random sample of adults from each state. The number of

adults sampled in each state is proportional to the population size of each state.

(d) The following scenario is an example of simple random sampling: An administrator at a University is interested in conducting a survey to investigate how students are adapting to online instruction.

The administrator decides to sample every 10th student from an alphabetical listing of all students that

attend the University.

(e) Observational studies can be used to infer cause-and-effect relationships between variables.

Question) Let X1, X2, · · · , Xn be independent and identically distributed (i.i.d.) random variables.

Let E(Xi) = µ and V ar(Xi) = σ

2

. Let T =

Pn

i=1 Xi

. Show that E(T) = nµ and V ar(T) = nσ2

.