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evelyng58
08.04.2020 •
Mathematics
A total of 9 cycles have been observed during a direct time study. The mean for the largest element time = 1.30 min, and the corresponding sample standard deviation s = 0.20 min. (a) Based on these data, what is the 95% confidence interval on the 1.30 min element time? (b) If the analyst wants to be 98% confident that the mean of the sample was within 5% of the true mean, how many more observations should be taken?
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Ответ:
a)
So on this case the 95% confidence interval would be given by (1.146;1.454)
b)![n=(\frac{2.326(0.2)}{0.065})^2 =51.22 \approx 52](/tpl/images/0588/8811/90470.png)
So the answer for this case would be n=52 rounded up to the nearest integer
Step-by-step 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".
s=0.2 represent the sample standard deviation
n=9 represent the sample size
Part a
The confidence interval for the mean is given by the following formula:
In order to calculate the critical value
we need to find first the degrees of freedom, given by:
Since the Confidence is 0.95 or 95%, the value of
and
, and we can use excel, a calculator or a table to find the critical value. The excel command would be: "=-T.INV(0.025,8)".And we see that ![t_{\alpha/2}=2.306](/tpl/images/0588/8811/61955.png)
Now we have everything in order to replace into formula (1):
So on this case the 95% confidence interval would be given by (1.146;1.454) Part b
The margin of error is given by this formula:
And on this case we have that ME =0.05*1.3= 0.065 and we are interested in order to find the value of n, if we solve n from equation (a) we got:
The critical value for 98% of confidence interval now can be founded using the normal distribution. And in excel we can use this formla to find it:"=-NORM.INV(0.01;0;1)", and we got
, replacing into formula (b) we got:
So the answer for this case would be n=52 rounded up to the nearest integer
Ответ: