A survey is being conducted in a city with 1 million residents. It would be far too expensiveto survey all of the residents, so a random sample of size 1000 is chosen (in practice, thereare many challenges with sampling, such as obtaining a complete list of everyone in the city,and dealing with people who refuse to participate). The survey is conducted by choosingpeople one at a time, with replacement and with equal probabilities.(a) Explain how sampling with versus without replacement here relates to thebirthday problem.(b) Find the probability that at least one person will get chosen more than once.

A survey is being conducted in a city with 1 million residents. It would be far too expensiveto survey all of the residents, so a random sample of size 1000 is chosen (in practice, thereare many challenges with sampling, such as obtaining a complete list of everyone in the city,and dealing with people who refuse to participate). The survey is conducted by choosingpeople one at a time, with replacement and with equal probabilities.(a) Explain how sampling with versus without replacement here relates to thebirthday problem.(b) Find the probability that at least one person will get chosen more than once.

A First Course in Probability (10th Edition)

10th Edition

ISBN:9780134753119

Author:Sheldon Ross

Publisher:Sheldon Ross

Chapter1: Combinatorial Analysis

Section: Chapter Questions

Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...

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12. Side effects Even if an experiment is double-blind, the blinding might be compromised if side effects of the treatments differ. For example, suppose researchers at a skin-care company are comparing their new acne treatment against the leading competitor. Fifty subjects are assigned at random to each treatment, and the company's researchers will rate the improvement for each of the 100 subjects. The researchers aren't told which subjects received which treatments, but they know that their new acne treatment causes a slight reddening of the skin. How might this knowledge compromise the blinding? Explain why this is an important consideration in this experiment.
3. Bayesian Inference A cab was involved at night in a hit and run accident. In this town there are only two cab companies, the Blue with 80% of the cabs and the Green company with the remaining 20%. As it happened there was a witness to the accident who identified the hit and run cab as Blue. The case came to trial. An important part of the testimony was a test of the reliability of the eyewitness to identify a cab as being either Blue or Green under the same circumstances as those that existed the night of the accident. The result of these tests was that the witness correctly identified a blue cab as blue 70% of the time and a green cab as green 80% of the time. On the basis of the claim of the eyewitness and the test conducted by the court, what is the probability that the cab involved in the accident was in fact Blue as the witness had claimed?
Suppose that Mairin has knit a collection of wool‑blend infinity scarves that she wants to sell in her Etsy shop for $28.00 each. However, she is worried that if the average price of comparable scarves is less than $28.00, hers would be less likely to sell. Mairin selected a random sample of 9 similar hand‑knit, wool‑blend infinity scarves, and their prices are as follows. $21.75,$24.00,$27.25,$29.95,$23.00,$26.00,$27.80,$33.50,$27.00$21.75,$24.00,$27.25,$29.95,$23.00,$26.00,$27.80,$33.50,$27.00 Use a left‑tailed one‑sample ?t‑test to determine whether the average price of hand‑knit, wool‑blend infinity scarves sold on Etsy is less than $28.00. Assume that the prices of all comparable scarves are normally distributed. Mairin should (reject, accept, fail to accept) her null hypothesis. There is (sufficient, no, insufficient) evidence that the mean price of hand‑knit, wool‑blend infinity scarves is (greater than, different from, less than, equal two) $28.00.
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A researcher studying public opinion of proposed Social Security changes obtains a simple random sample of 25 adult Americans and asks them whether or not support the proposed changes . To say that the distribution of the sample proportion of adults who respond yes, is approximately normal , how many more adult Americans does the researcher need to sample in the following cases? (a) 15% of all adult Americans support the changes (b) 20% of all adult Americans support the changes
2. In a population it is estimated that 20% have a desired trait of interest for the researcher. The researcher wants to know how many people on average he has to draw from a population to get 2 people with the trait. Use rows 20-24 of the Random Number Table to carry out the simulation. Explain clearly how you set up the problem and report your findings. Answer:
A researcher studying public opinion of proposed Social Security changes obtains a simple random sample of 50 adult Americans and asks them whether or not they support the proposed changes. To say that the distribution of the sample proportion of adults who respond yes, is approximately normal, how many more adult Americans does the researcher need to sample in the following cases? (a) 15% of all adult Americans support the changes (b) 20% of all adult Americans support the changes (a) The researcher must ask more American adults. (Round up to the nearest integer.) (b) The researcher must ask more American adults. (Round up to the nearest integer.)
Up-County Associates is hired to estimate the proportion of county households that own three or more automobiles. They conducted their survey using a simple random sample of 250 randomly selected households and found that 23% of those households owned three or more automobiles. (a) Is the proportion, 23%, a parameter or a statistic? Explain. (b) If Up-County Associates were to continually choose simple random samples of 250 randomly selected households, each time recording the proportion of households in the sample owning three or more automobiles, what would be the “shape” of the resulting distribution of all these sample proportions? Explain. (c) Suppose now that survey company Lower-County Polling also conducted a survey to determine the proportion of county households that own three or more automobiles, but they surveyed only 100 randomly selected households. If Lower-County Polling were also to sample over and over again (using simple random samples…