if the confidence interval increases, then what happens to the precision?
i. Three things influence the margin of error in a confidence interval gauge of a population mean: sample size, variability in the population, and conviction level. For each of these quantities separately, explain briefly what happens to the margin of error as that quantity increases.
Respond: As sample size increases, the margin of error decreases. As the variability in the population increases, the margin of fault increases. As the confidence level increases, the margin of mistake increases. Incidentally, population variability is not something we can usually control, but more meticulous collection of information can reduce the variability in our measurements. The third of these—the relationship betwixt conviction level and margin of error seems contradictory to many students considering they are confusing accuracy (confidence level) and precision (margin of error). If yous want to be surer of hitting a target with a spotlight, then y'all make your spotlight bigger.
2. A survey of 1000 Californians finds reports that 48% are excited by the almanac visit of INSPIRE participants to their fair state. Construct a 95% conviction interval on the true proportion of Californians who are excited to be visited by these Statistics teachers.
Respond: Nosotros beginning check that the sample size is large enough to use the normal approximation. The true value of p is unknown, so we can't check that np > ten and n(1-p) > 10, but nosotros can check this for p-hat, our estimate of p. m*.48 = 480 > x and k*.52 > 10. This means the normal approximation will be practiced, and nosotros can utilize them to calculate a confidence interval for p.
.48 +/- 1.96*sqrt(.48*.52/one thousand)
.48 +/- .03096552 (that mysterious 3% margin of error!)
(.45, .51) is a 95% CI for the true proportion of all Californians who are excited almost the Stats teachers' visit.
3. Since your interval contains values above 50% and therefore does finds that it is plausible that more than than half of the state feels this style, there remains a large question mark in your heed. Suppose you decide that y'all want to refine your estimate of the population proportion and cut the width of your interval in half. Volition doubling your sample size do this? How large a sample will exist needed to cut your interval width in half? How big a sample will exist needed to compress your interval to the point where 50% will non be included in a 95% conviction interval centered at the .48 point estimate?
Answer: The current interval width is about 6%. So the current margin of mistake is 3%. Nosotros desire margin of error = i.5% or
1.96*sqrt(.48*.52/due north) = .015
Solve for n: due north = (1.96/.015)^2 * .48*.52 = 4261.6
Nosotros'd need at least 4262 people in the sample. So to cut the width of the CI in half, we'd demand virtually four times as many people.
Assuming that the true value of p = .48, how many people would we need to brand certain our CI doesn't include .50? This means the margin of error must be less than two%, so solving for north:
n = (ane.96/.02)^2 *.48*.52 = 2397.one
We'd need nigh 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 hateful time it takes rats to exercise it. It is [2.3min, 3.1 min]. Which of the following statements is/are true? (More than ane statement may be correct.)
(A) 95% of the lab rats in the sample ran the maze in between ii.3 and 3.ane minutes.
(B) 95% of the lab rats in the population would run the maze in betwixt two.3 and 3.ane minutes.
(C) There is a 95% probability that the sample mean fourth dimension is between 2.3 and 3.ane minutes.
(D) There is a 95% probability that the population mean lies between ii.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 mean would be between 2.iii and 3.ane minutes.
(F) If I were to take many random samples of 67 lab rats and construct confidence intervals of maze-running time, well-nigh 95% of the time, the interval would contain the population mean. [2.3, iii.1] is the ane such possible interval that I computed from the random sample I actually observed.
(M) [2.3, 3.one] is the set of possible values of the population hateful maze-running fourth dimension that are consistent with the observed data, where "consequent" ways that the observed sample hateful falls in the centre ("typical") 95% of the sampling distribution for that parameter value.
Answer: F and G are both correct statements. None of the others are correct.
If you lot said (A) or (B), recall that nosotros are estimating a mean.
If you lot said (C), (D), or (E), remember that the interval [two.three, 3.1] has already been calculated and is not random. The parameter mu, while unknown, is non random. And then no statements can exist made about the probability that mu does anything or that [2.three, 3.1] does anything. The probability is associated with the random sampling, and thus the process that produces a confidence interval, non with the resulting interval.
5. Two students are doing a statistics project in which they drop toy parachuting soldiers off a building and try to get them to land in a hula-hoop target. They count the number of soldiers that succeed and the number of drops total. In a written report analyzing their data, they write the post-obit:
"We synthetic a 95% confidence interval estimate of the proportion of jumps in which the soldier landed in the target, and we got [0.l, 0.81]. We can exist 95% confident that the soldiers landed in the target between fifty% and 81% of the fourth dimension. Because the army desires an judge with greater precision than this (a narrower confidence interval) we would like to repeat the written report with a larger sample size, or echo our calculations with a higher confidence level."
How many errors can you spot in the in a higher place paragraph?
Answer: There are three wrong statements. First, the first argument should read "…the proportion of jumps in which soldiers land in the target." (Nosotros're estimating a population proportion.) 2nd, the second sentence besides refers to by tense and hence implies sample proportion rather than population proportion. It should read, "Nosotros can be 95% confident that soldiers state in the target between l% and 81% of the time." (The difference is subtle but shows a student misunderstanding.) And the third fault is in the final judgement. A college confidence level would produce a wider interval, non a narrower one.
Source: http://inspire.stat.ucla.edu/unit_10/solutions.php
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