Statistics: Confidence Intervals and Sample Size

This document discusses confidence intervals for mean and percentage in statistics. It explains how to calculate confidence intervals and provides examples. It also covers determining the additional sample size needed to meet specific requirements.

STATISTICS
[Document subtitle]
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Sample mean = $600
Sample standard deviation = $500
Number of samples = 400
(1) 95% confidence interval for the mean of all the claims processed by the insurance
company next year
The t value for 95% confidence interval = 1.9659
Standard error = Sample standard deviation/ sqrt (Number of samples) = 500/ sqrt (400) = 25
Margin of error = t value * Standard error = 1.9659*25 = 49.147
Lower limit of 95% confidence interval = Mean - Margin of error = 600 – 49.147 =550.85
Upper limit of 95% confidence interval = Mean +Margin of error= 600 +49.147 =649.15
95% confidence interval = [ 550.85 649.15]
It can be said with 95% confidence that mean of all the claims processed by the insurance
company next year would fall between 550.85 to 649.15.
(2) Percentage of the claims processed by the insurance company next year that would
exceed $1000 and also 95% confidence interval for this percentage
Percentage of the claims processed by the insurance company next year that would exceed
$1000 = (50+20+10)/(250+70+50+20+10) = 0.20 or 20%
95% confidence interval for this percentage is computed below.
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(3) Number of claims processed by processed by the insurance company next year that would
exceed $1000 and also 95% confidence interval for this number
Number of the claims processed by the insurance company next year that would exceed
$1000 = (50+20+10) = 80 (Assuming Mean)
95% confidence interval
Margin of error = t value * Standard error = 1.9*25 = 49.147
Lower limit of 95% confidence interval = Mean - Margin of error = 80 – 49.147 =30.85
Upper limit of 95% confidence interval = Mean +Margin of error= 80 +49.147 =129.15
95% confidence interval = [ 30.85 129.15]
It can be said with 95% confidence that mean number of all the claims processed by the
insurance company next year that would exceed $1000 would fall between 30.85 to 129.15.
(4) Sample size
(a) Number of additional claims that must be sampled to satisfy requirement I
The z value for 95% probability = 1.960
e=2 5
SS= ( z∗σ
e )
2
=( 1.96∗500
25 )
2
=1537
Additional claims = 1537 – 400 = 1137
(b) Number of additional claims that must be sampled to satisfy requirement II
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The z value for 95% probability = 1.960
e=2.5 %
Percentage of claims exceeding $1000 = 0.20
SS= z2 p( 1− p)
e2
SS= ( 1.96 )2∗( 0.20 )∗1−0.20
( 0.025 )2 =984
Additional claims = 984– 400 = 584
(c) Number of additional claims that must be sampled to satisfy both requirement
The additional claims must be 1137 in order to satisfy both the requirements.
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