David E. Brown is an expert in wildlife conservation. In his book The Wolf in the Southwest: The Making of an Endangered Species (University of Arizona Press), he records the following weights of adult grey wolves from two regions in Old Mexico.

Chihuahua region: x₁ variable in pounds

Durango region: x₂ variable in pounds

(a) Use a calculator with mean and standard deviation keys to calculate x₁, s₁, x₂, and s₂. (Use 2 decimal places.)

(b) Let μ₁ be the mean weight of the population of all grey wolves in the Chihuahua region. Let μ₂ be the mean weight of the population of all grey wolves in the Durango region. Find a 99% confidence interval for μ₁ – μ₂. (Use 2 decimal places.)

To evaluate the performance of inspectors in a new company, a quality manager had a sample of 12 novice inspectors evaluate 200 finished products. The same 200 items were evaluated by 12 experienced inspectors. SD of error for the novice inspectors was 8.64 and for the experiences inspectors was 5.74.

Was the variance in inspection errors lower for experienced inspectors than for novice inspectors? Conduct a hypothesis testing at  alpha = 0.05 and show all the work.

The number of men and women among professors in Math, Physics, Chemistry, Linguistics, and English departments from a SRS of small colleges were counted, and the results are shown in the table below.

Test the claim that the gender of a professor is independent of the department. Use the significance level α=0.01

(a) The test statistic is χ^2=

Suppose 130 geology students measure the mass of an ore sample. Due to human error and limitations in the reliability of the​ balance, not all the readings are equal. The results are found to closely approximate a normal​ curve, with mean

82g and standard deviation

1g. Use the symmetry of the normal curve and the empirical rule as needed to estimate the number of students reporting readings between 79

g and 85g.

The length of time to complete a door assembly on an automobile factory assembly line is normally distributed with mean μ=7.3 minutes and standard deviation σ=2 minutes. Samples of size 70 are taken. What is the mean value for the sampling distribution of the sample means?

The length of time to complete a door assembly on an automobile factory assembly line is normally distributed with mean μ=7.2 minutes and standard deviation σ=2.4 minutes. Samples of size 100 are taken. To the nearest thousandth of a minute, what is the standard deviation of the sampling distribution of the sample means?

Find P(46≤x¯≤53) for a random sample of size 35 with a mean of 51 and a standard deviation of 12.   (Round your answer to four decimal places.)

At a gymnastics meet, three judges evaluate the balance beam performances of five gymnasts. The judges use a scale of 1 to 10, where 10 is a perfect score. A statistician wants to examine the objectivity and consistency of the judges. Assume scores are normally distributed. (You may find it useful to reference the q table.)

a-1. Construct an ANOVA table. (Round intermediate calculations to at least 4 decimal places. Round "SS", "MS", "p-value" to 4 decimal places and "F" to 3 decimal places.)

a-2. If average scores differ by gymnast, use Tukey’s HSD method at the 5% significance level to determine which gymnasts’ performances differ. (Negative values should be indicated by a minus sign. Round intermediate calculations to at least 4 decimal places. Round your answers to 2 decimal places.)

1. Nobel is a serial gambler and regularly plays several rounds of a gamble in which he wins $10,000 if the head comes on top twice when a fair coin is flipped twice and loses $5,000 for any other outcome from the two flips of the coin. In other words, the outcome of the gamble is determined by flipping a coin twice on each round of the gamble and Nobel wins only if head comes on top on both flips of the coin and loses if any other outcome occurs. Nobel is considering to play 10 rounds of such a gamble next week hoping that he will win back the $20,000 he lost in a similar gamble last week. If we let X represent the number of wins for Nobel out of the next 10 rounds of the gamble, X can be assumed to have a binomial probability distribution. Please answer the following questions based on the information given above.

a. Please calculate the probability of success for Nobel on each round of the gamble. Show how you arrived at your answer.

b. What is the probability that Nobel will win none of the 10 rounds of the gamble?  

c. What is the probability that Nobel will lose more than 5 of the 10 rounds of the gamble?   

d. What is the probability that Nobel will win at least 8 out of the 10 rounds of the gamble?   

e. What is the probability that Nobel will lose less than 4 of the 10 rounds of the gamble?   

f. What is the probability that Nobel will lose no more than 7 out of the 10 rounds of the gamble? g. How much money is Nobel expected to win in the 10 rounds of the gamble? How much money is he expected to lose? Given your results, do you think Nobel is playing a smart gamble? Please show how you arrived at your results and explain your final answer.

h. Calculate and interpret the standard deviation for the number of wins for Nobel in the next 10 rounds of the gamble. Show your work.

2. Vehicles arrive at a toll bridge at an average rate of 180 an hour. Only one toll booth is currently open and can process arrivals (collect tolls) at a mean rate of 22 seconds per vehicle.

A. How many vehicles should be expected at the toll bridge in a 10-minute period? Please show how you arrived at your answer.

B. Define X to be the number of vehicles arriving at the toll bridge in a 10-minute period and assume that X has a Poisson Probability distribution. Please calculate the probability that 25 vehicles will arrive at the toll bridge in a 10-minute period.

C. Please calculate the probability of at least 20 vehicles arriving at the bridge within 10-minute period.                                                                  

D. Please calculate the probability of less than 28 vehicles arriving at the bridge within 10-minute period.

E. Please calculate the probability that no vehicle will arrive at the bridge within a 10-minute period.

F. Please calculate the probability of more than 35 vehicles arriving at the bridge within a 10-minute period. Show your work.   

G. Please calculate the probability of at most 22 vehicles arriving at the bridge within a 10-minute period. Show your work.

G. Calculate and interpret the standard deviation of X.   

I. For what purpose can the kind of information you have calculated above be used? If the state department of transportation wants to reduce the average wait time for the drivers to less than 25 seconds at the toll bridge, do you think they should open another toll booth at the bridge? Please explain.             

For this problem, carry at least four digits after the decimal in your calculations. Answers may vary slightly due to rounding.

The National Council of Small Businesses is interested in the proportion of small businesses that declared Chapter 11 bankruptcy last year. Since there are so many small businesses, the National Council intends to estimate the proportion from a random sample. Let p be the proportion of small businesses that declared Chapter 11 bankruptcy last year.

(a) If no preliminary sample is taken to estimate p, how large a sample is necessary to be 99% sure that a point estimate p̂ will be within a distance of 0.09from p? (Round your answer up to the nearest whole number.)
small businesses

(b) In a preliminary random sample of 30 small businesses, it was found that ten had declared Chapter 11 bankruptcy. How many more small businesses should be included in the sample to be 99% sure that a point estimate p̂ will be within a distance of 0.090 from p? (Round your answer up to the nearest whole number.)
more small businesses

Use R to do each of the following. Use R code instructions that are as general as possible, and also as efficient as possible. Use the Quick-R website for help on finding commands. 1. The following is a random sample of CT scores selected from 32 Miami students. 28, 27, 29, 27, 29, 31, 32, 30, 34, 30, 27, 25, 30, 32, 35, 32 23, 26, 27, 33, 33, 33, 31, 25, 28, 34, 30, 33, 28, 26, 30, 28 (a) Find the mean and standard deviation of this sample. Give the interpretation of the mean in the context. (b) Find the five numbers summary and provide the interpretation of the sample median in the context. (c) Draw the histogram of the data. What can you say about the distribution of the data. (d) Draw the Boxplot of the data. Is there any potential outlier? 2. The weekly amount spent for maintenance and repairs in a certain company has an approximately normal distribution, with a mean of $600 and a standard deviation of $40. If $700 is budgeted to cover repairs for next week, (a) what is the probability that the actual costs will exceed the budgeted amount? (b) how much should be budgeted weekly for maintenance and repairs to ensure that the probability that the budgeted amount will be exceeded in any given week is only 0.1?

When an automobile is stopped by a roving safety patrol, each tire is checked for tire wear, and each headlight is checked to see whether it is properly aimed. Let X denote the number of headlights that need adjustment, and let Y denote the number of defective tires.

(a) If X and Y are independent with p_X(0) = 0.5, p_X(1) = 0.3, p_X(2) = 0.2, and p_Y(0) = 0.1, p_Y(1) = 0.2, p_Y(2) = p_Y(3) = 0.05, p_Y(4) = 0.6, display the joint pmf of (X, Y) in a joint probability table.

(b) Compute P(X ≤ 1 and Y ≤ 1) from the joint probability table.
P(X ≤ 1 and Y ≤ 1) =  

Does P(X ≤ 1 and Y ≤ 1) equal the product P(X ≤ 1) · P(Y ≤ 1)?

YesNo    

(c) What is P(X + Y = 0) (the probability of no violations)?
P(X + Y = 0) =  

(d) Compute P(X + Y ≤ 1).
P(X + Y ≤ 1) =

Question 4. The probability that an individual will suffer from a bad reaction of a given serum is .003. Out of 10000 individuals, compute the following probabilities using both the Binomial and Poisson models.

Part 1. P (X = 0) where X is Binomial and then Poisson.

Part 2. P (X > 10) where X is Binomial and then Poisson.

In the Focus Problem at the beginning of this chapter, a study was described comparing the hatch ratios of wood duck nesting boxes. Group I nesting boxes were well separated from each other and well hidden by available brush. There were a total of 489 eggs in group I boxes, of which a field count showed about 266 hatched. Group II nesting boxes were placed in highly visible locations and grouped closely together. There were a total of 798 eggs in group II boxes, of which a field count showed about 262 hatched.

(a) Find a point estimate p̂1 for p1, the proportion of eggs that hatch in group I nest box placements. (Round your answer to three decimal places.) p̂1 = Find a 90% confidence interval for p1. (Round your answers to three decimal places.)

lower limit

upper limit

(b) Find a point estimate p̂2 for p2, the proportion of eggs that hatch in group II nest box placements. (Round your answer to three decimal places.) p̂2 = Find a 90% confidence interval for p2. (Round your answers to three decimal places.)

lower limit

upper limit

(c) Find a 90% confidence interval for p1 − p2. (Round your answers to three decimal places.)

lower limit

upper limit

Test the claim that for the population of statistics final exams, the mean score is 73 using alternative hypothesis that the mean score is different from 73. Sample statistics include n=18, x¯¯¯=74, and s=16. Use a significance level of α=0.01. (Assume normally distributed population.)

The test statistic is

The positive critical value is

The negative critical value is

The conclusion is A. There is sufficient evidence to reject the claim that the mean score is equal to 73. B. There is not sufficient evidence to reject the claim that the mean score is equal to 73.

A car service charges customers a flat fee per ride (which is higher during rush hour traffic) plus charges for each minute and each mile. Suppose that, in a certain metropolotian area during rush hour, the flat fee is $3, the cost per minute is $0.20, and the cost per mile is $1.20. Let x be the number of minutes and y the number of miles. At the end of a ride, the driver said that the passenger owed $20.60 and remarked that the number of minutes was five times the number of miles. Find the number of minutes and the number of miles for this trip.

A. Complete the equation that represents the total cost of the ride. ___=20.60

B. Complete the equation that represents the relationship between the number of minutes and number of miles. __=0

C. Find the number of minutes and number of miles for this trip.