Question Help 7.3.11 measured amounts of lead (in micrograms per cubic meter, or

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Question Help 7.3.11 measured amounts of lead (in micrograms per cubic meter, or g/m3) in the air. The EPA has established an Listed below are air quality standard for lead the given values to construct a 95% confidence interval estimate of the mean amount of lead in the air. Is there anything a this data set suggesting that the confidence interval might not be very good? of 1.5 g/m3 The measurements shown below were recorded at a building on different days. Use bout 540 1.00 0.49 0.77 0.74 1 00 Click here to view a t distribution table Click here to view page 1 of the standard normal distribution table. Click here to view page 2 of the standard normal distribution table What is the confidence interval for the population mean ? (Round to three decimal places as needed)

Explanation / Answer

TRADITIONAL METHOD
given that,
sample mean, x =1.5667
standard deviation, s =1.8875
sample size, n =6
I.
stanadard error = sd/ sqrt(n)
where,
sd = standard deviation
n = sample size
standard error = ( 1.8875/ sqrt ( 6) )
= 0.771
II.
margin of error = t /2 * (stanadard error)
where,
ta/2 = t-table value
level of significance, = 0.05
from standard normal table, two tailed value of |t /2| with n-1 = 5 d.f is 2.571
margin of error = 2.571 * 0.771
= 1.981
III.
CI = x ± margin of error
confidence interval = [ 1.5667 ± 1.981 ]
= [ -0.414 , 3.548 ]
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DIRECT METHOD
given that,
sample mean, x =1.5667
standard deviation, s =1.8875
sample size, n =6
level of significance, = 0.05
from standard normal table, two tailed value of |t /2| with n-1 = 5 d.f is 2.571
we use CI = x ± t a/2 * (sd/ Sqrt(n))
where,
x = mean
sd = standard deviation
a = 1 - (confidence level/100)
ta/2 = t-table value
CI = confidence interval
confidence interval = [ 1.5667 ± t a/2 ( 1.8875/ Sqrt ( 6) ]
= [ 1.5667-(2.571 * 0.771) , 1.5667+(2.571 * 0.771) ]
= [ -0.414 , 3.548 ]
-----------------------------------------------------------------------------------------------
interpretations:
1) we are 95% sure that the interval [ -0.414 , 3.548 ] contains the true population mean
2) If a large number of samples are collected, and a confidence interval is created
for each sample, 95% of these intervals will contains the true population mean

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