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# Sample Size and Probability

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1 ) Management has asked a lab to produce accurate results at the 88% level for a first time experiment. the management wants the final error to be within 1/10th of the sample standard deviation. Find the sample size.

2) A sample of 16 women had a mean score of 275 and a variance value of 360.
A. find both the 955 and 99% confidence intervals. what is the standard error and the maximum error for each?
B. Repeat "a" group of 28 women.
C. Explain the difference in the answers for both confidence levels and both sample size.

https://brainmass.com/statistics/confidence-interval/sample-size-probability-346736

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1)
For 88% confidence, z = 1.5548
E = s/8 which means s/E = 8
N = (z * s/E)^2
N = (1.5548 * 8)^2 = 154.71
Minimum sample size required is 155.

2)
(a) (i) 95% confidence interval:
Data:
n = 16
x-bar = 275
s = 18.9737
% = 95
Standard Error, SE = s/âˆšn = 4.7434
Degrees of freedom = 15
t- score = 2.1314
Maximum Error = Width of the confidence interval = t * SE = 10.1104
Lower Limit of the confidence interval = x-bar - width = 264.8896
Upper Limit of the confidence interval = x-bar + width = 285.1104
The confidence interval is [264.8896 285.1104]
(ii) 99% confidence interval:
Data:
n = 16
x-bar = 275
s = 18.9737
% = 99
Standard Error, SE = s/âˆšn = 4.7434
Degrees of freedom = 15
t- score = 2.9467
Maximum Error = Width of the confidence interval = t * SE = 13.9775
Lower Limit of the confidence interval = x-bar - width = 261.0225
Upper Limit of the confidence interval = x-bar + width = 288.9775
The confidence interval is [261.0225, 288.9775]
(b) (i) 95% confidence interval:
Data:
n = 28
x-bar = 275
s = 18.9737
% = 95
Standard Error, SE = s/âˆšn = 3.5857
Degrees of freedom = 27
t- score = 2.0518
Maximum Error = Width of the confidence interval = t * SE = 7.3572
Lower Limit of the confidence interval = x-bar - width = 267.6428
Upper Limit of the confidence interval = x-bar + width = 282.3572
The confidence interval is [267.6428, 282.3572]
(ii) 99% confidence interval:
Data:
n = 28
x-bar = 275
s = 18.9737
% = 99
Standard Error, SE = s/âˆšn = 4.7434
Degrees of freedom = 15
t- score = 2.7707
Maximum Error = Width of the confidence interval = t * SE = 9.9348
Lower Limit of the confidence interval = x-bar - width = 265.0652
Upper Limit of the confidence interval = x-bar + width = 284.9348
The confidence interval is [265.0652, 284.9348]
(c) We observe that as the confidence level increases, the maximum error of estimate also increases (width of the confidence interval increases). As the sample size increases, the maximum error of estimate decreases (width of the confidence interval decreases).

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