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?

a)    

So on this case the 95% confidence interval would be given by (1.146;1.454)  

b)

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".

represent the sample mean

population mean (variable of interest)

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:

  (1)

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

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:

   (a)

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:

  (b)

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

step-by-step explanation:

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step-by-step explanation: