A common technique for estimating animal populations is to tag and release individual animals in two different outings. This procedure is called catch and release. If the wildlife remain in the sampling area and are randomly caught, a fraction of the animals tagged during the first outing are likely to be caught again during the second outing. Based on the number tagged and the fraction caught twice, the total number of animals in the area can be estimated. a. Consider a case in which $200$ fish are tagged and released during the first outing. During a second outing in the same area, $200$ fish are again caught and released, of which one-half are already tagged. Estimate $N$, the total number of fish in the entire sampling area. Explain your reasoning, b. Consider a case in which $200$ fish are tagged and released during the first outing. During a second outing in the same area, $200$ fish are again caught and released, of which one-fourth are already tagged. Estimate $N$ the total number of fish in the entire sampling area. Explain your reasoning. c. Generalize your results from parts (a) and (b) by letting $p$ be the fraction of tagged fish that are caught during the second outing. Find a formula for the function $N=f(p)$ that relates the total number of fish, $N$, to the fraction tagged during the second outing, $p$. d. Graph the function obtained in part (c). What is the domain? Explain. e. Suppose that $15 \%$ of the fish in the second sample are tagged. Use the formula from part (c) to estimate the total number of fish in the sampling area. Confirm your result on your graph. f. Locate a real study in which the catch and release method was used. Report on the specific details of the study and describe how closely it followed the theory outlined in this problem.

Decide whether the sampling method could result in biased sample. Explain your reasoning.

A write for a travel magazine wants to learn tourists' opinions about the nightlife in a city. She decides to visit some of the tourist attractions in the city on a Thursday morning. She plants to ask the first 50 tourists she meets for their opinions.

A company maintains three offices in a certain region, each staffed by two employees. Information concerning yearly salaries (1000s of dollars) is as follows:

$$\begin{array}{lcccccc} \text {Office} & 1 & 1 & 2 & 2 & 3 & 3 \\ \text {Employee} & 1 & 2 & 3 & 4 & 5 & 6 \\ \text {Salary} & 29.7 & 33.6 & 30.2 & 33.6 & 25.8 & 29.7 \end{array}$$

a. Suppose two of these employees are randomly selected from among the six (without replacement). Determine the sampling distribution of the sample mean salary $\bar{X}$. b. Suppose one of the three offices is randomly selected. Let $X_{1}$ and $X_{2}$ denote the salaries of the two employees. Determine the sampling distribution of $\bar{X}$. c. How does $E(\bar{X})$ from parts (a) and (b) compare to the population mean salary $\mu$?

The Food Marketing Institute shows that $17\%$ of households spend more than $100$ per week on groceries. Assume the population proportion is $p = .17$ and a simple random sample of $800$ households will be selected from the population.

Show the sampling distribution of $\bar{p}$, the sample proportion of households spending more than $100$ per week on groceries.

The ski patrol at Vail, Colorado, is studying slab avalanches in its region. A random sample of avalanches in spring gave the following thicknesses (in cm):

$$\begin{array}{ccccccc} 59 & 51 & 76 & 38 & 65 & 54 & 49 & 62 \\ 68 & 55 & 64 & 67 & 63 & 74 & 65 & 79 \end{array}$$

i. Use a calculator with mean and standard deviation keys to verify that $\bar{x} \approx 61.8$ and $s \approx 10.6 \mathrm{cm}$. ii. Assume the slab thickness has an approximately normal distribution. Use a 1% level of significance to test the claim that the mean slab thickness in the Vail region is different from that in Canada. What sampling distribution will you use? Explain the rationale for your choice of sampling distribution. Compute the value of the sample test statistic.

An automaker has found that the lifetime of its batteries varies from car to car according to a normal distribution with mean $\mu=48$ months and standard deviation $\sigma=8.2$ months. The company installs a new battery on an SRS of 8 cars. Calculate the mean and standard deviation of the sampling distribution of $\bar{x}$ for SRSs of size 8.

An urn contains 5 red, 6 blue, and 8 green balls. If a set of 3 balls is randomly selected, what is the probability that each of the balls will be (a) of the same color? (b) of different colors? Repeat under the assumption that whenever a ball is selected, its color is noted and it is then replaced in the urn before the next selection. This is known as sampling with replacement.

Following are descriptions of behavior that are characteristic among users of certain classes of drugs. For each description, indicate the class of drug (narcotics, stimulants, and so on) for which the behavior is most characteristic. For each description, also name at least one drug that produces the described effects.
a) Slurred speech, slow reaction time, impaired judgment, reduced coordination
b) Intense emotional responses, anxiety, altered sensory perceptions
c) Alertness, feelings of strength and confidence, rapid speech and movement, decreased appetite
d) Drowsiness, intense feeling of well-being, relief from pain

A random sample of size $n=50$ is taken from a population with mean $\mu=-9.5$ and standard deviation $\sigma=2$.

a. Calculate the expected value and the standard error for the sampling distribution of the sample mean.

b. What is the probability that the sample mean is less than $-10 ?$

c. What is the probability that the sample mean falls between $-10$ and $-9$ ?

The Hodges-Lehmann shift estimate is defined to be $\hat{\Delta}=\operatorname{median}\left(X_{i}-Y_{j}\right)$, where $X_{1}, X_{2}, \ldots, X_{n}$ are independent observations from a distribution F and $Y_{1}, Y_{2}, \ldots, Y_{m}$ are independent observations from a distribution G and are independent of the $X_{i}$. a. Show that if F and G are normal distributions, then $E(\hat{\Delta})=\mu_{X}-\mu_{Y}$. b. Why is $\hat{\Delta}$ robust to outliers? c. What is $\hat{\Delta}$ for the previous problem and how does it compare to the differences of the means and of the medians? d. Use the bootstrap to approximate the sampling distribution and the standard error of $\hat{\Delta}$. e. From the bootstrap approximation to the sampling distribution, form an approximate 90% confidence interval for $\hat{\Delta}$.

When 3 births are randomly selected, the sample space is bbb, bbg, bgb, bgg, gbb, gbg, ggb, and ggg. Assume that those 8 outcomes are equally likely. Describe the sampling distribution of the proportion of girls from 3 births as a probability distribution table. Does the mean of the sample proportions equal the proportion of girls in 3 births?

You know at least four different strategies for checking to see if two algebraic expressions are equivalent or not-comparing tables or graphs of $(x, y)$. values, comparing the reasoning that led from the problem conditions to the expressions, or using algebraic reasoning based on number system properties like the distributive property.

a. What do you see as the advantages and disadvantages of each strategy?

b. How do you decide which strategy to use in a given situation?

c. Which strategy gives you the most confidence in your judgment about whether the given expressions are or are not equivalent?

Tell whether the sampling strategy will result in a random sample from the population under consideration. If it is not random, explain why it is not. The HR director of a large law firm wants to survey a random sample of the firm's 326 employees. She numbers the employees alphabetically and uses her daughter's calculator to choose 50 distinct numbers with the command "randlnt (1,326.50)."

Suppose the probability is p = .2 that a person who purchases an instant lottery ticket wins money, and this probability holds for every ticket purchased. Consider different random samples of n = 64 purchased tickets. Let $\hat{p}$ = sample proportion of winning tickets in a sample of 64 tickets.

Give the numerical value of the mean of the sampling distribution of $\hat{p}$