Confidence Intervals

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Solutionsto Practice Problems

1. Three things influence the margin of error in aconfidence interval estimate of a population mean: sample dimension,varicapacity in the population, and confidence level. For each of thesequantities independently, explain briefly what happens to the margin oferror as that amount boosts.

Answer: As sample size rises, the margin oferror decreases. As the variability in the population boosts, themargin of error rises. As the confidence level boosts, themargin of error boosts. Incidentally, populace variability is notsomepoint we deserve to typically manage, yet more meticulous collection ofdata deserve to mitigate the varicapacity in our dimensions. The 3rd ofthese—the connection in between confidence level and also margin of errorappears contradictory to many type of students bereason they are confusingaccuracy (confidence level) and also precision (margin of error). If youdesire to be surer of hitting a targain through a spotlight, then you makeyour spotlight bigger.

2. A survey of 1000 Californians finds reports that 48% areexcited by the yearly visit of participants to their fairsteustatiushistory.orge. Construct a 95% confidence interval on the true propercentage ofCalifornians that are excited to be saw by these steustatiushistory.orgisticsteachers.

Answer: We initially inspect that the sample sizeis big enough to apply the normal approximation. The true value of pis unrecognized, so we can not inspect that np > 10 and also n(1-p) > 10, butwe deserve to inspect this for p-hat, our estimate of p. 1000*.48 = 480 > 10and also 1000*.52 > 10. This means the normal approximation will certainly be good,and also we deserve to use them to calculate a confidence interval for p.

.48 +/- 1.96*sqrt(.48*.52/1000)

.48 +/- .03096552 (that mysterious 3% margin of error!)

(.45, .51) is a 95% CI for the true propercent of allCalifornians that are excited about the steustatiushistory.orgs teachers" visit.

3. Since your interval has values over 50% andtherefore does finds that it is plausible that more than half of thesteustatiushistory.orge feels this means, tbelow remains a large question mark in your mind.Suppose you decide that you want to refine your estimate of thepopulation proportion and also reduced the width of your interval in half. Willdoubling your sample size carry out this? How big a sample will certainly be needed toreduced your interval width in half? How big a sample will certainly be required toshrink your interval to the suggest wright here 50% will certainly not be consisted of in a95% confidence interval focused at the .48 allude estimate?

Answer: The existing interval width is about6%. So the existing margin of error is 3%. We desire margin of error =1.5% or

1.96*sqrt(.48*.52/n) = .015

Solve for n: n = (1.96/.015)^2 * .48*.52 = 4261.6

We"d need at leastern 4262 people in the sample. So to reduced thewidth of the CI in fifty percent, we"d need around 4 times as many type of people.

Assuming that the true worth of p = .48, exactly how many type of civilization wouldwe have to make certain our CI doesn"t include .50? This suggests the marginof error have to be less than 2%, so resolving for n:

n = (1.96/.02)^2 *.48*.52 = 2397.1

We"d require around 2398 civilization.

4. A random sample of 67 lab rats are enticed to run througha maze, and a 95% confidence interval is constructed of the intend timeit takes rats to carry out it. It is <2.3min, 3.1 min>. Which of the followingsteustatiushistory.orgements is/are true? (More than one steustatiushistory.orgement may be correct.)

(A) 95% of the lab rats in the sample ran the maze in between 2.3 and3.1 minutes.(B) 95% of the lab rats in the population would run the maze in between2.3 and 3.1 minutes.(C) Tbelow is a 95% probcapacity that the sample suppose time is between 2.3and 3.1 minutes.(D) There is a 95% probability that the population intend lies between2.3 and 3.1 minutes.(E) If I were to take many kind of random samples of 67 lab rats and also takesample implies of maze-running times, around 95% of the moment, the samplesuppose would certainly be in between 2.3 and 3.1 minutes.(F) If I were to take many random samples of 67 lab rats and constructconfidence intervals of maze-running time, around 95% of the moment, theinterval would certainly contain the populace intend. <2.3, 3.1> is the one suchfeasible interval that I computed from the random sample I actuallyobserved.(G) <2.3, 3.1> is the collection of feasible values of the population meanmaze-running time that are consistent through the oboffered data, where“consistent” indicates that the oboffered sample intend falls in the middle(“typical”) 95% of the sampling distribution for that parameter value.

Answer: F and G are both correct steustatiushistory.orgements.Namong the others are correct.

If you shelp (A) or (B), remember that we are estimating aintend.

If you shelp (C), (D), or (E), remember that the interval<2.3, 3.1> has actually currently been calculated and is not random. The parametermu, while unwell-known, is not random. So no steustatiushistory.orgements deserve to be made aboutthe probcapacity that mu does anypoint or that <2.3, 3.1> does anything.The probcapability is associated via the random sampling, and for this reason theprocess that produces a confidence interval, not with the resultinginterval.

5. Two students are doing a steustatiushistory.orgistics task in which theydrop toy parachuting soldiers off a building and try to obtain them toland also in a hula-hoop target. They count the number of soldiers thatsucceed and the variety of drops total. In a report analyzing theirinformation, they create the following:“We built a 95% confidence interval estimate of the proportion ofjumps in which the soldier landed in the taracquire, and also we gained <0.50,0.81>. We can be 95% confident that the soldiers landed in the targetin between 50% and 81% of the moment. Because the army desires an estimatevia higher precision than this (a narrower confidence interval) wewould favor to repeat the study through a larger sample dimension, or repeat ourcalculations via a greater confidence level.”How many errors have the right to you spot in the above paragraph?

Answer: There are three incorrect steustatiushistory.orgements.First, the initially steustatiushistory.orgement should review “…the proportion of jumps inwhich soldiers land also in the targain.” (We’re estimating a populationpropercentage.) Second, the second sentence likewise refers to past tense andthus suggests sample propercent fairly than populace propercentage.

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Itmust check out, “We deserve to be 95% confident that soldiers land in the targetin between 50% and also 81% of the time.” (The distinction is subtle however mirrors astudent misknowledge.) And the third error is in the last sentence.A greater confidence level would certainly create a broader interval, not anarrower one.