How Does Increasing The Level Of Confidence Affect The Size Of The Margin Ofã¢â‚¬â€¹ Error, E?

 Solutions to Practise Problems

i. Three things influence the margin of error in a confidence interval estimate of a population mean: sample size, variability in the population, and confidence level. For each of these quantities separately, explain briefly what happens to the margin of mistake as that quantity increases.

Answer: As sample size increases, the margin of error decreases. As the variability in the population increases, the margin of fault increases. As the conviction level increases, the margin of error increases. Incidentally, population variability is non something we tin can usually control, but more meticulous drove of information can reduce the variability in our measurements. The third of these—the relationship between confidence level and margin of error seems contradictory to many students because they are confusing accuracy (confidence level) and precision (margin of error). If you desire to exist surer of hitting a target with a spotlight, then you lot make your spotlight bigger.

2. A survey of 1000 Californians finds reports that 48% are excited by the annual visit of INSPIRE participants to their fair state. Construct a 95% confidence interval on the true proportion of Californians who are excited to be visited past these Statistics teachers.

Answer: We kickoff cheque that the sample size is large enough to apply the normal approximation. The truthful value of p is unknown, so we can't cheque that np > 10 and north(one-p) > 10, but nosotros can cheque this for p-hat, our estimate of p. 1000*.48 = 480 > ten and 1000*.52 > x. This means the normal approximation volition be adept, and we tin can use them to calculate a conviction interval for p.

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

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

(.45, .51) is a 95% CI for the true proportion of all Californians who are excited well-nigh the Stats teachers' visit.

iii. Since your interval contains values to a higher place 50% and therefore does finds that it is plausible that more than than half of the country feels this fashion, at that place remains a big question mark in your mind. Suppose you decide that you lot want to refine your estimate of the population proportion and cut the width of your interval in half. Will doubling your sample size do this? How large a sample will be needed to cut your interval width in one-half? How large a sample will be needed to shrink your interval to the betoken where l% volition not be included in a 95% confidence interval centered at the .48 signal judge?

Respond: The current interval width is about 6%. And then the current margin of error is 3%. We want margin of fault = 1.v% or

one.96*sqrt(.48*.52/northward) = .015

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

We'd need at to the lowest degree 4262 people in the sample. And then to cutting the width of the CI in half, we'd need about 4 times as many people.

Assuming that the truthful value of p = .48, how many people would we demand to make sure our CI doesn't include .fifty? This means the margin of error must exist less than 2%, and so solving for due north:

northward = (i.96/.02)^2 *.48*.52 = 2397.1

Nosotros'd need virtually 2398 people.

4. A random sample of 67 lab rats are enticed to run through a maze, and a 95% confidence interval is constructed of the mean time information technology takes rats to do it. It is [2.3min, 3.1 min]. Which of the following statements is/are truthful? (More than one statement may exist right.)

(A) 95% of the lab rats in the sample ran the maze in between 2.3 and 3.1 minutes.
(B) 95% of the lab rats in the population would run the maze in between 2.3 and 3.one minutes.
(C) There is a 95% probability that the sample mean fourth dimension is betwixt 2.iii and three.one minutes.
(D) There is a 95% probability that the population hateful lies between two.3 and 3.ane minutes.
(East) If I were to take many random samples of 67 lab rats and take sample means of maze-running times, about 95% of the time, the sample hateful would be betwixt 2.three and 3.1 minutes.
(F) If I were to take many random samples of 67 lab rats and construct confidence intervals of maze-running fourth dimension, well-nigh 95% of the time, the interval would contain the population mean. [2.3, three.1] is the i such possible interval that I computed from the random sample I actually observed.
(G) [2.3, iii.i] is the gear up of possible values of the population mean maze-running fourth dimension that are consistent with the observed data, where "consistent" means that the observed sample mean falls in the heart ("typical") 95% of the sampling distribution for that parameter value.

Reply: F and G are both right statements. None of the others are right.

If you said (A) or (B), remember that we are estimating a mean.

If y'all said (C), (D), or (E), remember that the interval [two.3, three.1] has already been calculated and is not random. The parameter mu, while unknown, is non random. And so no statements tin can be fabricated nigh the probability that mu does anything or that [two.3, 3.1] does anything. The probability is associated with the random sampling, and thus the process that produces a conviction interval, not with the resulting interval.

v. 2 students are doing a statistics project in which they drop toy parachuting soldiers off a building and try to go them to land in a hula-hoop target. They count the number of soldiers that succeed and the number of drops total. In a report analyzing their data, they write the following:
"We synthetic a 95% confidence interval approximate of the proportion of jumps in which the soldier landed in the target, and we got [0.l, 0.81]. We can be 95% confident that the soldiers landed in the target between 50% and 81% of the time. Considering the army desires an judge with greater precision than this (a narrower conviction interval) we would like to echo the study with a larger sample size, or echo our calculations with a higher confidence level."
How many errors can yous spot in the above paragraph?

Answer: At that place are iii incorrect statements. First, the commencement statement should read "…the proportion of jumps in which soldiers land in the target." (We're estimating a population proportion.) 2d, the second sentence too refers to past tense and hence implies sample proportion rather than population proportion. It should read, "We can exist 95% confident that soldiers state in the target betwixt 50% and 81% of the time." (The difference is subtle but shows a student misunderstanding.) And the 3rd error is in the concluding sentence. A college conviction level would produce a wider interval, not a narrower 1.

How Does Increasing The Level Of Confidence Affect The Size Of The Margin Ofã¢â‚¬â€¹ Error, E?,

Source: http://inspire.stat.ucla.edu/unit_10/solutions.php

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