How Does Sample Size Affect Margin Of Error

 Solutions to Exercise Problems

1. 3 things influence the margin of fault in a confidence interval approximate 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 error as that quantity increases.

Reply: As sample size increases, the margin of error decreases. As the variability in the population increases, the margin of error increases. As the confidence level increases, the margin of mistake increases. Incidentally, population variability is not something nosotros can commonly control, but more than meticulous collection of data tin can reduce the variability in our measurements. The 3rd of these—the relationship between confidence level and margin of error seems contradictory to many students because they are disruptive accuracy (confidence level) and precision (margin of fault). If you want to be surer of hitting a target with a spotlight, then you lot make your spotlight bigger.

ii. A survey of 1000 Californians finds reports that 48% are excited by the almanac visit of INSPIRE participants to their off-white state. Construct a 95% confidence interval on the true proportion of Californians who are excited to be visited by these Statistics teachers.

Answer: Nosotros first check that the sample size is large enough to apply the normal approximation. The truthful value of p is unknown, so we tin't bank check that np > 10 and n(i-p) > ten, but we can bank check this for p-hat, our approximate of p. g*.48 = 480 > 10 and chiliad*.52 > ten. This means the normal approximation will be skilful, and we can use them to summate a confidence interval for p.

.48 +/- i.96*sqrt(.48*.52/1000)

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

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

three. Since your interval contains values to a higher place 50% and therefore does finds that information technology is plausible that more than half of the state feels this fashion, there remains a big question mark in your mind. Suppose you decide that you desire to refine your estimate of the population proportion and cutting the width of your interval in half. Will doubling your sample size do this? How large a sample will be needed to cutting your interval width in half? How big a sample volition exist needed to shrink your interval to the indicate where fifty% will non be included in a 95% confidence interval centered at the .48 point estimate?

Respond: The current interval width is about 6%. So the current margin of error is 3%. We want margin of error = 1.five% or

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

Solve for n: n = (ane.96/.015)^2 * .48*.52 = 4261.6

Nosotros'd need at least 4262 people in the sample. Then to cut the width of the CI in half, we'd need nigh four times as many people.

Assuming that the true value of p = .48, how many people would we need to make certain our CI doesn't include .50? This means the margin of fault must be less than 2%, so solving for northward:

due north = (one.96/.02)^2 *.48*.52 = 2397.i

We'd need nearly 2398 people.

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

(A) 95% of the lab rats in the sample ran the maze in betwixt 2.iii and 3.1 minutes.
(B) 95% of the lab rats in the population would run the maze in between 2.3 and 3.1 minutes.
(C) There is a 95% probability that the sample hateful fourth dimension is betwixt 2.3 and 3.1 minutes.
(D) In that location is a 95% probability that the population mean lies betwixt ii.3 and 3.1 minutes.
(Due east) If I were to take many random samples of 67 lab rats and take sample ways of maze-running times, about 95% of the time, the sample mean would be between 2.3 and 3.i minutes.
(F) If I were to take many random samples of 67 lab rats and construct confidence intervals of maze-running time, near 95% of the time, the interval would contain the population mean. [ii.3, iii.1] is the 1 such possible interval that I computed from the random sample I actually observed.
(K) [ii.3, iii.1] is the set of possible values of the population mean maze-running time that are consequent with the observed data, where "consistent" means that the observed sample mean falls in the middle ("typical") 95% of the sampling distribution for that parameter value.

Answer: F and Thou are both correct statements. None of the others are correct.

If you said (A) or (B), recall that nosotros are estimating a hateful.

If you said (C), (D), or (Due east), call up that the interval [2.3, 3.one] has already been calculated and is not random. The parameter mu, while unknown, is not random. And then no statements tin can be fabricated about the probability that mu does anything or that [2.3, iii.ane] does anything. The probability is associated with the random sampling, and thus the process that produces a confidence interval, non with the resulting interval.

five. 2 students are doing a statistics project in which they driblet toy parachuting soldiers off a building and try to become them to land in a hula-hoop target. They count the number of soldiers that succeed and the number of drops full. In a written report analyzing their data, they write the following:
"We constructed a 95% confidence interval estimate of the proportion of jumps in which the soldier landed in the target, and we got [0.50, 0.81]. We can exist 95% confident that the soldiers landed in the target between l% and 81% of the time. Because the army desires an estimate with greater precision than this (a narrower confidence interval) nosotros would like to repeat the written report with a larger sample size, or repeat our calculations with a higher conviction level."
How many errors tin you spot in the to a higher place paragraph?

Reply: In that location are three incorrect statements. Commencement, the outset statement should read "…the proportion of jumps in which soldiers country in the target." (We're estimating a population proportion.) 2d, the 2nd sentence also refers to past tense and hence implies sample proportion rather than population proportion. It should read, "We tin be 95% confident that soldiers land in the target between l% and 81% of the time." (The divergence is subtle but shows a student misunderstanding.) And the third error is in the last judgement. A higher confidence level would produce a wider interval, not a narrower one.

How Does Sample Size Affect Margin Of Error,

Source: http://inspire.stat.ucla.edu/unit_10/solutions.php#:~:text=Answer%3A%20As%20sample%20size%20increases,the%20margin%20of%20error%20increases.

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