Please solve and show step by step solution Ten (10) samples of the heating valu
ID: 3299636 • Letter: P
Question
Please solve and show step by step solution
Ten (10) samples of the heating value (kJ/kg) of natural gas from a certain gas field are measured as follows: 48530, 48980, 50210, 49860, 48560, 49540, 49270, 48850, 49320, 48680 (a) Calculate for a 95% confidence level the random uncertainty of each measurement (b) Calculate the random uncertainty of the mean of the measurements (c) Assume, a large number of samples (N > 30) were taken and it has the same standard deviation S as the above 10 samples, calculate the random uncertainty of the large number of measurementsExplanation / Answer
TRADITIONAL METHOD
given that,
sample mean, x =49180
standard deviation, s =566.3136
sample size, n =10
I.
stanadard error = sd/ sqrt(n)
where,
sd = standard deviation
n = sample size
standard error = ( 566.3136/ sqrt ( 10) )
= 179.084
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 = 9 d.f is 2.262
margin of error = 2.262 * 179.084
= 405.088
III.
CI = x ± margin of error
confidence interval = [ 49180 ± 405.088 ]
= [ 48774.912 , 49585.088 ]
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DIRECT METHOD
given that,
sample mean, x =49180
standard deviation, s =566.3136
sample size, n =10
level of significance, = 0.05
from standard normal table, two tailed value of |t /2| with n-1 = 9 d.f is 2.262
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 = [ 49180 ± Z a/2 ( 566.3136/ Sqrt ( 10) ]
= [ 49180-(2.262 * 179.084) , 49180+(2.262 * 179.084) ]
= [ 48774.912 , 49585.088 ]
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interpretations:
1) we are 95% sure that the interval [ 48774.912 , 49585.088 ] 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
[ANSWERS]
a. [ 48774.912 , 49585.088 ]
b. mean, x =49180