Question

You are told that a 95% CI for expected lead content when traffic flow is 15, based on a sample of n = 12 observations, is (461.2, 598.6). Calculate a CI with confidence level 99% for expected lead content

asked Aug 19, 2021 235k views

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Jonas Anso · Answer 1 · 2021-08-24T16:41:58+0000

Answer:

$\bar X = (461.2+598.6)/(2)=529.9$

ME= (Width)/(2)= (137.4)/(2)= 68.7

The margin of error is given by:

$ME = t_(\alpha/2) SE$

For 95% of confidence the value of the significance is
$\alpha =0.05$ and
$\alpha/2 = 0.025$ , the degrees of freedom are given by:

df = 12-1 = 11

And then we can calculate the critical value for 95% with df = 11 and we got
$t_(\alpha)= 2.20$

And then we can find the standard error:

$SE = (ME)/(t_(\alpha/2))= (68.7)/(2.20)= 31.227$

$t_(\alpha/2)= 3.11$

And using the confidence interval formula we got:

$\bar X \pm t_(\alpha/2) SE$

529.9 - 3.11*31.227 = 432.784

529.9 + 3.11*31.227 = 627.016

Explanation:

Previous concepts

A confidence interval is "a range of values that’s likely to include a population value with a certain degree of confidence. It is often expressed a % whereby a population means lies between an upper and lower interval".

The margin of error is the range of values below and above the sample statistic in a confidence interval.

Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".

$\bar X$ represent the sample mean for the sample

$\mu$ population mean (variable of interest)

s represent the sample standard deviation

n represent the sample size

Solution to the problem

The confidence interval for the mean is given by the following formula:

$\bar X \pm t_(\alpha/2)(s)/(√(n))$ (1)

For the 95% confidence interval we know that the result is given by (461.2, 598.6), we can estimate the sample mean like this:

$\bar X = (461.2+598.6)/(2)=529.9$

Because the distribution is symmetrical, now we can estimate the width of the interval like this:

Width = 598.6-461.2= 137.4

And the margin of error is given by: