5 astronomy teams from across the world measured the distance to Proxima Centaur
ID: 3311694 • Letter: 5
Question
5 astronomy teams from across the world measured the distance to Proxima Centauri using a new method. It is reasonable to assume that the error of this method is normally distributed, but, since it is new, there is no information about its standard variation. Find a 95%-confidence interval for the distance if the obtained mea- surements are (in light years) 4.20, 4.16,4.04,4.01,4.20. What would your confidence interval look like if they used an established method whose standard deviation of the measurement error is 0.1?Explanation / Answer
Q5.
TRADITIONAL METHOD
given that,
sample mean, x =4.122
standard deviation, s =0.0907
sample size, n =5
I.
stanadard error = sd/ sqrt(n)
where,
sd = standard deviation
n = sample size
standard error = ( 0.0907/ sqrt ( 5) )
= 0.041
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 = 4 d.f is 2.776
margin of error = 2.776 * 0.041
= 0.113
III.
CI = x ± margin of error
confidence interval = [ 4.122 ± 0.113 ]
= [ 4.009 , 4.235 ]
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DIRECT METHOD
given that,
sample mean, x =4.122
standard deviation, s =0.0907
sample size, n =5
level of significance, = 0.05
from standard normal table, two tailed value of |t /2| with n-1 = 4 d.f is 2.776
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 = [ 4.122 ± t a/2 ( 0.0907/ Sqrt ( 5) ]
= [ 4.122-(2.776 * 0.041) , 4.122+(2.776 * 0.041) ]
= [ 4.009 , 4.235 ]
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interpretations:
1) we are 95% sure that the interval [ 4.009 , 4.235 ] 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
when the standard deviation has the measurment error the calculated value of the interval would narrower or wider
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