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.)

(c) Examine the confidence interval and explain what it means in the context of this problem. Does the interval consist of numbers that are all positive? all negative? of different signs? At the 99% level of confidence, what can you say about the comparison of the average weight of grey wolves in the Chihuahua region with the average weight of grey wolves in the Durango region?

Because the interval contains only positive numbers, we can say that the mean weight of grey wolves is greater in the Chihuahua region.Because the interval contains only negative numbers, we can say that the mean weight of grey wolves is greater in the Durango region.    Because the interval contains both positive and negative numbers, we can not say that the mean weight of grey wolves is greater in the Chihuahua region.We can not make any conclusions using this confidence interval.

Application Exercise:
An ecologist hypothesizes that birds with longer wing spans use wider tree branches. The ecologist captured male birds, measured their wing lengths and other characteristics in millimeters, and then marked and released them. During the ensuing winter, the ecologist repeatedly observed the marked birds as they foraged for food on tree branches. He noted the branch diameter on each occasion, and calculated the average branch diameter for each bird in centimeters. The measurement data are below. What can the ecologist conclude with an α of 0.01?

a) What is the appropriate statistic?
---Select--- na Correlation Slope Chi-Square
Compute the statistic selected above:  

c) Compute the appropriate test statistic(s) to make a decision about H₀.
(Hint: Make sure to write down the null and alternative hypotheses to help solve the problem.)
critical value =  ; test statistic =  
Decision:  ---Select--- Reject H0 Fail to reject H0

d) Compute the corresponding effect size(s) and indicate magnitude(s).
If not appropriate, input and/or select "na" below.
effect size =  ;   ---Select--- na trivial effect small effect medium effect large effect

e) Make an interpretation based on the results.

There was a significant positive relationship between wing length of the marked birds and branch diameter.

There was a significant negative relationship between wing length of the marked birds and branch diameter.     

There was no significant relationship between wing length of the marked birds and branch diameter.

a) The flow in a river can be modeled as a log-normal distribution. From the data, it was estimated that, the probability that the flow exceeds 855 cfs is 50% and the probability that it exceeds 100 cfs is 90%. Let X denote the flow in cfs in the river. Flood conditions occur when flow is 5000 cfs or above. To compute the percentage of time flood conditions occur for this river, we have to find, P(X≥5000)=1-P(Z<a). What is the value of a? Please report your answer in 3 decimal places.

b) What is the probability of P(Z<2.23)? Please report your answer in 3 decimal places.

What information is provided by the numerical value of the Pearson correlation? Also in your own words, explain in detail and include a discussion of X, Y, co-variability and separate variability in your answer.

Suppose we are interested in bidding on a piece of land and we know one other bidder is interested. The seller announced that the highest bid in excess of $10,000 will be accepted. Assume that the competitor's bid x is a random variable that is uniformly distributed between $10,000 and $15,400.

1. Suppose you bid $12,000. What is the probability that your bid will be accepted (to 2 decimals)?
   
2. Suppose you bid $14,000. What is the probability that your bid will be accepted (to 2 decimals)?
   
3. What amount should you bid to maximize the probability that you get the property (in dollars)?
   
4. Suppose that you know someone is willing to pay you $16,000 for the property. You are considering bidding the amount shown in part (c) but a friend suggests you bid $13,000. If your objective is to maximize the expected profit, what is your bid?
   Select Stay with your bid in part (c); it maximizes expected profit or Bid $13000 to maximize the expected profit
   
   What is the expected profit for this bid (in dollars)?

The International Air Transport Association surveys business travelers to develop quality ratings for transatlantic gateway airports. The maximum possible rating is 10. Suppose a simple random sample of 50 business travelers is selected and each traveler is asked to provide a rating for the Miami International Airport. The ratings obtained from the sample of 50 business travelers follow.

Develop a 95% confidence interval estimate of the population mean rating for Miami (to 2 decimals).

( , )

The unique colors of the cashmere sweaters your firm makes result from heating undyed yarn in a kettle with a dye liquor. The pH (acidity) of the liquor is critical for regulating dye uptake and hence the final color. There are 5 kettles, all of which receive dye liquor from a common source. Past data show that pH varies according to a Normal distribution with μ = 4.69 and σ = 0.118. You use statistical process control to check the stability of the process. Twice each day, the pH of the liquor in each kettle is measured, giving a sample of size 5. The mean pH x is compared with "control limits" given by the 99.7 part of the 68−95−99.7 rule for normal distributions, namely

μ_x ± 3σ_x.

What are the numerical values of these control limits for x? (Round your answers to three decimal places.)

Use Excel to compute the descriptive statistics for the following data set:

25, 45, 73, 16, 34, 98, 34, 45, 26, 2, 56, 97, 12, 445, 23, 63, 110, 12, 17, and 41.

Answer the questions below using the appropriate statistical technique. For questions involving the use of hypothesis testing, you must:

1. State the null and research hypotheses

2. Provide the Z(critical), T(critical), or χ 2 (critical) score corresponding to the α threshold for your test

3. Provide your test statistic

4. Provide your decision about statistical significance

An advantage that often comes with a basic knowledge of statistics is a change in salary. To see whether this was the case for Tulane University graduates, you took a random sample of 57 students who completed a statistics class and asked about their starting salaries (in thousands) after graduation. The sample had a mean of 53.3 with a standard deviation of 3.72 (i.e., x = 53.3 and s = 3.72). A call to the Office of the Registrar indicates that the average starting salary value for all Tulane students is 47.1. Do students who take statistics courses earn an equal salary compared to Tulane students generally? Use α = 0.001.

Professor Fair believes that extra time does not improve grades on exams. He randomly divided a group of 300 students into two groups and gave them all the same test. One group had exactly 1 hour in which to finish the test, and the other group could stay as long as desired. The results are shown in the following table. Test at the 0.01 level of significance that time to complete a test and test results are independent.

(ii) Find the sample test statistic. (Round your answer to two decimal places.)

(iii) Find or estimate the P-value of the sample test statistic.

P-value > 0.1000

.050 < P-value < 0.100     

0.025 < P-value < 0.0500

.010 < P-value < 0.0250

.005 < P-value < 0.010

P-value < 0.005

The housing market has recovered slowly from the economic crisis of 2008.​ Recently, in one large​ community, realtors randomly sampled 29 bids from potential buyers to estimate the average loss in home value. The sample showed the average loss was ​$9440 with a standard deviation of ​$2030. In​ 2011, the average home in this region of the country lost ​$8874 in value. Was the community studied by the realtors​ unusual? Use a​ t-test to decide if the average loss observed was significantly different from the regional average with 0.05 as the​ P-value cutoff level.

An article in Journal of the American Statistical Association (1990, Vol. 85, pp. 972–985) measured weight of 30 rats under experiment controls. Suppose that there are 12 underweight rats. (a) Calculate a 90% two-sided confidence interval on the true proportion of rats that would show underweight from the experiment. Round your answers to 3 decimal places. Enter your answer; confidence interval, lower bound ≤p≤ Enter your answer; confidence interval, upper bound (b) Using the point estimate of p obtained from the preliminary sample, what sample size is needed to be 90% confident that the error in estimating the true value of p is no more than 0.02? n= Enter your answer in accordance to the item b) of the question statement (c) How large must the sample be if we wish to be at least 90% confident that the error in estimating p is less than 0.02, regardless of the true value of p? n= Enter your answer in accordance to the item c) of the question statement

X₁,...,X_n are i.i.d Γ(α₁,β),...,Γ(α_n,β) respectively. Show S = X₁ +···+X_n ∼ Γ(α₁ +···,α_n,β).

The Wechsler IQ test has a normal shape with μ = 100 and σ = 15. Describe the expected characteristics of the distribution of sample means for samples of n = 45. Mean = Standard error Shape = Still speaking of the Wechsler IQ test, if you take a random sample of n = 36 students, what is the probability that the sample mean will be between 96 and 104? Be careful about which formula you use. Still speaking of the Wechsler IQ test, if you take a random sample of n = 144 students, what is the probability that the sample mean will be between 96 and 104?