The number of telephone calls coming into the central switchboard of an office building follows a Poisson process with an average of 4 calls per minute.

a) What is the probability that at least 3 calls will arrive in a given one minute period?

b) What is the probability that at least 20 calls will arrive in a given five minute period?

c) What is the probability that the time from the opening of the switchboard until the first call exceeds 20 seconds?

d) What is the probability that the time from the opening of the switchboard until the 4th call is at most 45 seconds?

e) Find the mean and variance for the waiting time between the opening of the switchboard and the 3rd call.

Thickness measurements of ancient prehistoric Native American pot shards discovered in a Hopi village are approximately normally distributed, with a mean of 5.4 millimeters (mm) and a standard deviation of 0.8 mm. For a randomly found shard, find the following probabilities. (Round your answers to four decimal places.) (a) the thickness is less than 3.0 mm (b) the thickness is more than 7.0 mm (c) the thickness is between 3.0 mm and 7.0 mm

A hairdresser collected data on the number of services provided on Saturdays and on weekdays. Her results are listed below. Assume the two samples were independently taken. Saturday: n1=60 and x1=24 Weekday: n2=45 and x2=15

a) Estimate the difference in the true proportions with a 99% confidence interval.

b) Interpret this interval, in sense what can you conclude about the true difference between these two proportions.

An exhaustive study of all active Facebook accounts was recently conducted by Facebook. One variable Facebook recorded was the number of friends X of each Facebook user. Suppose X has expected value E(X) = 180 and standard deviation SD(X) = 280.7. Since the possible values of X are only integers and since the distribution of X is highly skewed to the right, the distribution of X cannot be described by a normal model. Suppose you select a random sample of 35 Facebook users and record the number of Facebook friends each user has.

A) What is the probability that the 35 Facebook users in your sample have a sample mean number of friends x greater than 177?

(use 4 decimal places in your answer)

B) What is the probability that the 35 Facebook users in your sample have a sample mean number of friends x less than 183?

(use 4 decimal places in your answer)

The heights of European 13-year-old boys can be approximated by a normal model with mean μ of 63.1 inches and standard deviation σ of 2.42 inches.

A) What is the probability that a randomly selected 13-year-old boy from Europe is taller than 65.1 inches?

(use 4 decimal places in your answer)

B) A random sample of 4 European 13-year-old boys is selected. What is the probability that the sample mean height x is greater than 65.1 inches?

(use 4 decimal places in your answer)

C) A random sample of 9 European 13-year-old boys is selected. What is the probability that the sample mean height x is greater than 65.1 inches?

(use 4 decimal places in your answer)

Discussion

How to use Excel to compute the descriptive statistics and/or calculate probabilities for a Normal/Binomial distribution?

You want to test whether salaries have gone up compared to in the year 2000. The 2000 census shows that the average annual salary of all Americans in 2000 was $35,152. You take an SRS of 10 Americans, ask for their annual salary for this year, and get the following results in thousands of dollars.

Annual Salary

45, 16, 24, 83, 103, 31, 69, 94, 72, 20

a) Determine the population(s) and parameter(s) being discussed

b) Determine which tool will help us find what we need (one sample z test, one sample t test, two sample t test, one sample z interval, two sample t intervals)

c) Check if the conditions for this tool hold

d) Whether or not the conditions hold, use the tool you chose in part (b) . Use C=95% for all confidence intervals and a=5% for all significance tests.

**Be sure that all methods end with a sentence describing the results**

Instructions:

What is the meaning of mutually exclusive events? What is meant by collectively exhaustive? In your responses, please give an example of each. Please also comment on the examples provided by other students in your responses to others.

Construct a box plot from the given data. Use the approximation method. Diameters of Cans in an Assembly Line: 5.7,5.2,5.7,5.6,5.1,5.3,5.5,5.3,5.6 Answer Draw the box plot by selecting each of the five movable parts to the appropriate position.

The body mass index for a sample of men and a sample of women are given below. assume the samples are simple random samples obtained form populations with normal distributions.

Men: 24.9, 28.4, 23.2, 32.8, 25.3, 32.9, 28.1, 28.6, 32.9, 26.4

Women: 19.6, 23.7, 18.2, 34.5, 19.3, 24.3, 18.8, 33.8, 18.7, 24.1

A. construct a 90% confidence interval estimate of the standard deviation of BMIs for men

_ < σmen < _

B. construct a 99% confidence interval estimate of the standard deviation of BMIs for women

_ < σwomen < _

In 1955, John Wayne played Genghis Khan in a movie called The Conqueror. The movie was filmed downwind of the site of eleven previous above-ground nuclear bomb tests. Of the 220 people who worked on the movie, 91 were diagnosed with cancer by the early 1980s, including Wayne, his co-stars, and the director. According to a large-scale epidemiological study, only ~14% of the general populace of similar age develop cancer. (Assume this probability relates to each of the movie people). Is there evidence for an increased cancer risk for people associated with the movie.

What is the best estimate of the probability of a movie person getting cancer within the study interval? What is the standard error of your estimate? What does this quantity measure? What is the 95% confidence interval for this probability estimate? Does this interval bracket the typical cancer rate of 14% from the large-scale study of the general populace? Interpret the result.

A study of the total number of minutes spent in the local emergency room revealed that patients spent on average 162.8 minutes in the ER, with a standard deviation of 53.4 minutes. Assume that the times follow a normal distribution. Let X be the processing time of a randomly selected patient.

(a) Describe the probability distribution of X and state its parameters μ and σ.

(b) Find the probability that the processing time of a randomly selected patient is

i. less than 3 hours.

ii. between 140 and 200 minutes.

iii. more than 2 hours.

(c) Find the 80-th percentile for the processing time of a randomly selected patient.

A process using a p-chart with L=3 has an in-control fraction defective of p=0.01. Sample size is 300. What is the probability of detecting a shift to an out-of-control value of p=0.05 for the first time on the second sample following the shift?