Question 1184849: A laboratory tested 78 chicken eggs and found that the mean amount of cholesterol was 224 milligrams with s = 14.7 milligrams. Find the 95% confidence interval.
Round your answer to the nearest whole number unless indicated otherwise.
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! sample size = 78
mean of sample = 224
standard deviation of sample = 14.7
assuming that this is a sample taken out of a much larger quantity of chicken eggs, you would do the following.
standard error = sample standard deviation divided by square root of sample size.
that's called s in this problem.
s = 14.7 / sqrt(78) = 1.6644 rounded to 4 decimal places.
since you are using the sample standard deviation, you would use the t-score, rather than the z-score.
the number of degrees of freedom is equal to the sample size minus 1 = 77.
since that's greater than 30, your t-score will probably be close to your z-score.
i'll do both to show you the difference.
the standard error is the same in both cases.
at 95% two-tailed confidence interval, the critical z-score is equal to plus or minus 1.960 rounded to 3 decimal places.
at 95% two-tailed confidence interval, the critical t-score, with 77 degrees of freedom, is equal to plus or minus 1.991 rounded to 3 decimal places.
they're not the same, but they're not all that different from each other.
the greater the sample size, the closer the t-score will get to the z-score.
the bigger difference is at smaller sample size.
assuming t-score, then the critical t-score formula will be:
t(77) = (x - m) / s
s is the raw score
m is the mean
s is the standard error
at the low end, this becomes:
-1.991 = (x - 224) / 1.6644 = 220.687 rounded to 3 decimal places.
at the high end, this becomes:
1.991 = (x - 224) / 1.6644 = 227.314 rounded to 3 decimal places.
assuming z-score, then the critical z-score formula will be:
z = (x - m) / s
at the low end, this becomes:
-1.960 = (x - 224) / 1.6644 = 220.738 rounded to 3 decimal places.
at the high end, this becomes:
1.960 = (x - 224) / 1.6644 = 227.262 rounded to 3 decimal places.
your confidence interval with t-score is 220.687 to 227.314.
your confidence interval with z-score is 220.738 to 227.262.
they're different, but not all that different.
with a greater sample size, the t-score gets closer to the z-score.
for practical purposes, with a sample size of 78, you could have used either.
the t-score is supposedly the more accurate, because it takes into account that the standard deviation of the sample will not always be the same as the standard deviation of the population, so there is some variability in there that is taken into account by the t-score formula.
your solution is the t-score results because you were using the sample standard deviation, rather than the population standard deviation.
here's a simple guide as to when to use each.
https://www.statisticshowto.com/probability-and-statistics/hypothesis-testing/t-score-vs-z-score/
since you did not have the population standard deviation, the table says to use the t-score, regardless of the sample size.
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