6. A lawyer commutes daily from his suburban home to his midtown office. The average time for a one-way trip is 24 minutes, with a standard deviation of 3.8 minutes. Assume the distribution of the trip-length to be normally distributed. (a) If the office opens at 9:00am and he leaves his house at 8:40 am daily, what percentage of the time is he late for work? You must draw the distribution and indicate the relevant numbers etc. You must also give the answer as a number. (b) Find the length of time above which we find the longest 20% of the trips. You must draw the distribution and indicate the relevant numbers etc. You must also give the answer as a number. (c) During a period of 20 work days, on how many days should you expect the lawyer to be late for work? (d) What is the probability that he is late on at most 10 of those 20 days?

The state of California has a mean annual rainfall of 27.6 inches, whereas the state of New York has a mean annual rainfall of 48.7 inches. Assume the standard deviation for California is 7.4 inches and for New York is 3.1 inches. Find the probability that, for a sample of 45 years of rainfall for California, the mean annual rainfall is at least 29 inches.

Let x represent the dollar amount spent on supermarket impulse buying in a 10-minute (unplanned) shopping interval. Based on a certain article, the mean of the x distribution is about $32 and the estimated standard deviation is about $7.

(a) Consider a random sample of n = 80 customers, each of whom has 10 minutes of unplanned shopping time in a supermarket. From the central limit theorem, what can you say about the probability distribution of x, the average amount spent by these customers due to impulse buying? What are the mean and standard deviation of the x distribution?

The sampling distribution of x is approximately normal with mean μ_x = 32 and standard error σ_x = $0.09. The sampling distribution of x is not normal.     The sampling distribution of x is approximately normal with mean μ_x = 32 and standard error σ_x = $7. The sampling distribution of x is approximately normal with mean μ_x = 32 and standard error σ_x = $0.78.

Is it necessary to make any assumption about the x distribution? Explain your answer.

It is necessary to assume that x has an approximately normal distribution. It is necessary to assume that x has a large distribution.     It is not necessary to make any assumption about the x distribution because μ is large. It is not necessary to make any assumption about the x distribution because n is large.

(b) What is the probability that x is between $30 and $34? (Round your answer to four decimal places.)

(c) Let us assume that x has a distribution that is approximately normal. What is the probability that x is between $30 and $34? (Round your answer to four decimal places.)
(d) In part (b), we used x, the average amount spent, computed for 80 customers. In part (c), we used x, the amount spent by only one customer. The answers to parts (b) and (c) are very different. Why would this happen?

1. Listed below are the budgets​ (in millions of​ dollars) and the gross receipts​ (in millions of​ dollars) for randomly selected movies. .

Find the value of the linear correlation coefficient r.

2. For a sample of eight​ bears, researchers measured the distances around the​ bears' chests and weighed the bears. Calculator was used to find that the value of the linear correlation coefficient is

r equals=0.963

What proportion of the variation in weight can be explained by the linear relationship between weight and chest​ size?

a. What proportion of the variation in weight can be explained by the linear relationship between weight and chest​ size?

3. Assume that you have paired values consisting of heights​ (in inches) and weights​ (in lb) from 40 randomly selected men. The linear correlation coefficient r is

0.559.

Find the value of the coefficient of determination. What practical information does the coefficient of determination​provide?

4.

The data show the bug chirps per minute at different temperatures. Find the regression​ equation, letting the first variable be the independent​ (x) variable. Find the best predicted temperature for a time when a bug is chirping at the rate of 3000 chirps per minute. Use a significance level of 0.05. What is wrong with this predicted​ value?

What is the regression​ equation?

5.

The data below shows height​ (in inches) and pulse rates​ (in beats per​ minute) of a random sample of women. .

Find the value of the linear correlation coefficient r.

r equals =

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

The state of California has a mean annual rainfall of 27.6 inches, whereas the state of New York has a mean annual rainfall of 44.5 inches. Assume the standard deviation for California is 6.3 inches and for New York is 8.2 inches. Find the probability that, for a sample of 40 years of rainfall for California, the mean annual rainfall is at least 29 inches.

Five people on the basement of a building get on an elevator that stops at seven floors. Assuming that each has an equal probability of going to any floor, find

(a) the probability that they all get off at different floors; (3 POINTS)

(b) the probability that two people get off at the same floor and all others get off at different floors. (4 POINTS)

Let’s consider a study that followed a randomly selected group of 100 State U students during a two-year period at the school. The study found that a linear relationship exists between the number of hours students spend engaging in social media each week and their cumulative gpa during the two-year period. The model for this relationship can be given by the equation

g?pa = −0.032 × (hours) + 2.944 (a) Interpret the slope of the line in the context of the data.

(b) The residual gpa for a particular student who spent 20 hours per week using social media was found to be 0.476. What was this student’s cumulative gpa during the two-year period?

(c) Would the correlation coefficient for the linear relationship be positive or negative? Explain.

(d) If another study found that the linear correlation coefficient between a student’s gpa and the number of hours spent at the library was r = 0.46, could you conclude that this relationship is stronger than the one between gpa and hours spent on social media? Explain.

Consider the following data collected from a sample of 12 American black bears:

(a) Sketch a scatterplot of the data. Treat length as the explanatory variable. Describe the association.

(b) Construct the equation for the line of best fit.

(c) Estimate the weight of a bear which measures 142.5 cm in length.

(d) What percent of the variation in the bears’ weights can be described by the differences in their lengths?

Assuming that the sample is representative of the population of all visitors to the website, is there evidence in the data at the 0.10 level of significance to conclude that the proportion of customers who use Chrome as their browser is less than 34%? Use hypothesis testing to answer your question. Put your hypotheses in the Answer. Put your p-value, rounded to at least 4 decimal places. In the Answer, state whether you reject or don’t reject the null hypothesis. state what you conclude about the proportion of customers who use Chrome as their browser.

Identify the sampling method for each scenario.

1. (a) An orchard farmer subdivides his orchard into 40 parcels of land. He randomly selects 4 parcels and inspects every tree on those parcels for the presence of a particular fungus.
2. (b) A librarian inspects every fifth book on a shelf starting with the second to determine the number of items missing bar codes.
3. (c) A hotel leaves customer satisfaction cards in each guest room. At the end of the week, the general manager reads all 60 cards that were filled out by guests that week.
4. (d) The College of Engineering randomly selects samples from each of its four branches (mechanical, civil, chemical, electrical) to assess the effectiveness of a new program for student advising.

In the following problem, check that it is appropriate to use the normal approximation to the binomial. Then use the normal distribution to estimate the requested probabilities. It is estimated that 3.5% of the general population will live past their 90th birthday. In a graduating class of 781 high school seniors, find the following probabilities. (Round your answers to four decimal places.) (a) 15 or more will live beyond their 90th birthday (b) 30 or more will live beyond their 90th birthday (c) between 25 and 35 will live beyond their 90th birthday (d) more than 40 will live beyond their 90th birthday

Truckloads of waste contaminated with cadmium may either be sent to a sanitary landfill or a hazardous waste landfill. The levels of cadmium vary daily so each morning a sample of truckloads must be tested to determine which landfill to use. If the mean level of cadmium exceeds the allowable amount of 1 milligram per liter for a sanitary landfill then the trucks must be sent further to the hazardous landfill. Of course, driving a further distance requires more expense for the company. You are in charge of testing and determining which landfill to use for dumping for today.

You will use a 5% level of significance for your hypothesis test.

1. State the null and alternative hypotheses, and identify which represents the claim. Why?

2. Determine the type of test that you should use: left-tailed, right-tailed, or two-tailed. Explain your reasoning.

3. What sampling technique would you use to determine your sample of truckloads that will be tested this morning? Should a small sample or a large sample be used? Does it really matter?

4. What decision concerning your null hypothesis would result in a Type I error? What is the interpretation and the implication of this error? What about a Type II error? Obviously, you want to minimize the risk of both types of error when decision making. Which of these errors is more serious in this situation?

5. Suppose that the null hypothesis is rejected when you perform your hypothesis test. Assuming that a correct decision was made, what do you believe regarding the mean level of cadmium on today's truckloads? Give a complete interpretation.

Problem:

A university administrator is interested in whether a new building can be planned and built on campus within a four-year time frame. He considers the process in two phases. Phase I: Phase I involves lobbying the state legislature and governor for permission and funds, issuing bonds to obtain funds, and obtaining all the appropriate legal documents. Past experience indicates that the time required to complete phase I is approximately normally distributed with a mean of 16 months and standard deviation of 4 months. If X = phase I time, then X ~ N(μ = 16 months, σ = 4 months). Phase II: Phase II involves creation of blue prints, obtaining building permits, hiring contractors, and, finally, the actual construction of the building. Past data indicates that the time required to complete these tasks is approximately normally distributed with a mean of 18 months and a standard deviation of 12 months. If Y = phase II time, then Y ~ N(μ = 18 months, σ = 12 months). a) A new random variable, T = total time for completing the entire project, is defined as T = X + Y. What is the probability distribution of T? (Give both the name of the distribution and its parameters.) b) Find the probability that the total time for the project is less than four years. (In symbols, calculate P(T < 48 months).) c) Find the 95th percentile of the distribution of T.

A person's blood glucose level and diabetes are closely related. Let x be a random variable measured in milligrams of glucose per deciliter (1/10 of a liter) of blood. Suppose that after a 12-hour fast, the random variable x will have a distribution that is approximately normal with mean μ = 89 and standard deviation σ = 22.

Note: After 50 years of age, both the mean and standard deviation tend to increase. For an adult (under 50) after a 12-hour fast, find the following probabilities. (Round your answers to four decimal places.)

(a) x is more than 60

(b) x is less than 110

(c) x is between 60 and 110

(d) x is greater than 125 (borderline diabetes starts at 125)