1. How many times do we need to toss the coin, keeping track of the number of tosses and the number of success, to ensure that P(0.95p ≤ pˆ ≤ 1.05p) ≥ 0.95? Please justify your answer as fully as you can.Note: That is, find an n that ensures that the probability that ˆp is within 5% of p is at least 0.95. p ofSuppose we have a coin with a probability p of coming up heads and q 1coming up tails on any given toss (so the coin is biased unless p = 0.5), but we are notgiven what p or q are. We know, from class or the textbook, that the expected value ofthe number of heads in n tosses is E(X) = np. If we repeatedly flip the coin and recordthe results, the number of heads that actually turn up, call it Ê, can be divided by thenumber of tosses, n, to give an estimate, ô = Ê/n, of p. The problem is that we're likelyto have to be pretty lucky for p = p, so the real question is how likely it is that p is closeto p. (This sort of thing is what statistics is all about: try infer from actual data what isreally true, and also provide an estimate of the likelihood that what is inferred is close toreality.)Let us suppose,for the sake of argument, that we know [somehow!] that 0.4 ≤p ≤ 0.6(so 0.4 < q = 1- p ≤ 0.6) for our possibly biased coin and that we have the time andpatience to toss it any finite number of times n that we need to. Further, let us suppose thatwe desire to get an estimate of p that is within 5% of the real value, i.e. 0.95p ≤ p ≤ 1.05p.

The following solution is provided below

1. How many times do we need to toss the coin, keeping track of the number of tosses and the number of success, to ensure that P(0.95p ≤ pˆ ≤ 1.05p) ≥ 0.95? Please justify your answer as fully as you can. Note: That is, find an n that ensures that the probability that ˆp is within 5% of p is at least 0.95.

1. How many times do we need to toss the coin, keeping track of the number of tosses and the number of success, to ensure that P(0.95p ≤ pˆ ≤ 1.05p) ≥ 0.95? Please justify your answer as fully as you can. Note: That is, find an n that ensures that the probability that ˆp is within 5% of p is at least 0.95.

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...

See similar textbooks

Similar questions

Hospitals typically require backup generators to provide electricity in the event of a power outage. Assume that emergency backup generators fail 44% of the times when they are needed. A hospital has two backup generators so that power is available if one of them fails during a power outage. Complete parts (a) and (b) below. a. Find the probability that both generators fail during a power outage. (Round to four decimal places as needed.) b. Find the probability of having a working generator in the event of a power outage. Is that probability high enough for the hospital? Assume the hospital needs both generators to fail less than 1% of the time when needed.(Round to four decimal places as needed.)
The probability of a number cube landing on 5 is 1/6. If a number cube is tossed 72 times, how many times can it be expected to land on 5?
Hospitals typically require backup generators to provide electricity in the event of a power outage. Assume that emergency backup generators fail 42% of the times when they are needed. A hospital has two backup generators so that power is available if one of them fails during a power outage. Complete parts (a) and (b) below. a. Find the probability that both generators fail during a power outage. (Round to four decimal places as needed.) b. Find the probability of having a working generator in the event of a power outage. Is that probability high enough for the hospital? Assume the hospital needs both generators to fail less than 1% of the time when needed. (Round to four decimal places as needed.) Is that probability high enough for the hospital? Select the correct answer below and, if necessary, fill in the answer box to complete your choice. O A. No, because both generators fail about % of the time they are needed. Given the importance of the hospital's needs, the reliability should…
Hospitals typically require backup generators to provide electricity in the event of a power outage. Assume that emergency backup generators fail 21% of the times when they are needed. A hospital has two backup generators so that power is available if one of them fails during a power outage. Complete parts (a) and (b) below. a. Find the probability that both generators fail during a power outage. (Round to four decimal places as needed.) b. Find the probability of having a working generator in the event of a power outage. Is that probability high enough for the hospital? Assume the hospital needs both generators to fail less than 1% of the time when needed. |(Round to four decimal places as needed.) Is that probability high enough for the hospital? Select the correct answer below and, if necessary, fill in the answer box to complete your choice. O A. Yes, because both generators fail about % of the time they are needed, which is low enough to not impact the health of patients. (Round…
Find the probability of the indicated event if P(E) = 0.20 and P(F) = 0.40. Find P(E or F) if P(E and F) = 0.20. P(E or F) = (Simplify your answer.)
According to a 2009 Reader's Digest article, people throw away about 11% of what they buy at the grocery store. Assume this is the true proportion and you plan to randomly survey 79 grocery shoppers to investigate their behavior. What is the probability that the sample proportion does not exceed 0.18?Note: Calculate z to two decimal places.Answer = (Enter your answer as a number accurate to 4 decimal places.)
Find the indicated probability and interpret the result. From 1975 through 2020, the mean annual gain of the Dow Jones Industrial Average was 653. A random sample of 34 years is selected from this population. What is the probability that the mean gain for the sample was between 400 and 800? Assume o = 1540. The probability is (Round to four decimal places as needed.) Interpret the result. Select the correct choice and fill in the answer box to complete your choice. (Round to two decimal places as needed.) O A. About OB. About OC. About OD. About % of samples of 46 years will have an annual mean gain between 400 and 800. % of samples of 34 years will have an annual mean gain between 653 and 800. % of samples of 34 years will have an annual mean gain between 400 and 800. % of sampl of 34 years will have an annual mean gain between 400 and 653.
The mean percent of childhood asthma prevalence in 43 cities is 2.38%. A random sample of 30 of these cities is selected. What is the probability that the mean childhood asthma prevalence for the sample is greater than 2.8%? Interpret this probability. Assume that o = 1.25%. The probability is (Round to four decimal places as needed.) Interpret this probability. Select the correct choice below and fill in the answer box to complete your choice. (Round to two decimal places as needed.) O A. About % of samples of 43 cities will have a mean childhood asthma prevalence greater than 2.8%. O B. About % of samples of 30 cities will have a mean childhood asthma prevalence greater than 2.38%. O C. About % of samples of 30 cities will have a mean childhood asthma prevalence greater than 2.8%. Click to select your answer(s). P Type here to search 99+ a (? 1:42 PM 4/11/2021 ho insen 144 ese & t backspace %23 3. %24 4 6. 8 2. Y W tab H. J. K pase 5 R
For each probability and percentile problem, draw the picture.A random number generator picks a number from 1 to 8 in a uniform manner. Part (h) P(X > 5 | X > 3) = (Enter an exact number as an integer, fraction, or decimal.) Part (i) Find the 60th percentile. (Round your answer to one decimal place.)
You have just stepped into the tub to take a shower when one of your contact lenses falls out. (You have not turned the shower on yet.) Assuming that the lens is equally likely to land anywhere on the bottom of the tub, what is the probability (expressed as a PERCENT to the nearest TENTH) that it landed in the drain? Use the dimensions shown in the image below. Diagram is not drawn to scale. 36 in. - Drain: 4 in. in diameter 30 in.