Families USA , a monthly magazine that discusses issues related to health and he
ID: 2843097 • Letter: F
Question
Families USA, a monthly magazine that discusses issues related to health and health costs, surveyed 19 of its subscribers. It found that the annual health insurance premiums for a family with coverage through an employer averaged $10,230. The standard deviation of the sample was $1,060.
Based on this sample information, develop a 95% confidence interval for the population mean yearly premium. (Round up your answers to the next whole number.)
How large a sample is needed to find the population mean within $278 at 95% confidence? (Round up your answer to the next whole number.)
(a)Based on this sample information, develop a 95% confidence interval for the population mean yearly premium. (Round up your answers to the next whole number.)
Explanation / Answer
a. ANSWER: 90% Resulting Confidence Interval for 'true mean': = [10599, 11359]
Why???
SMALL SAMPLE, CONFIDENCE INTERVAL, NORMAL POPULATION DISTRIBUTION
x-bar: Sample mean = 10979
s: Sample standard deviation = 1000
n: Number of samples = 20
df: degrees of freedom = 19
Confidence Level = 90
"Look-up" Table ('t-critical value') = 1.7
Look-up Table of ('t critical values') for confidence and prediction intervals. Central two-sided area = 90% with df = 19. Another Look-up method is to utilize Microsoft Excel function: TINV(probability,degrees_freedom) Returns the inverse of the Student's t-distribution 90% Resulting Confidence Interval for 'true mean': x-bar +/- ('t critical value') * s/SQRT(n) = 10979 +/- 1.7 * 1000/SQRT(20) = [10599, 11359]
b. ANSWER: Sample Size = 107 for 99% level of confidence
Why???
SMALL SAMPLE, LEVEL OF CONFIDENCE, NORMAL POPULATION DISTRIBUTION
Margin of Error (half of confidence interval) = 250
The margin of error is defined as the "radius" (or half the width) of a confidence interval for a particular statistic.
Level of Confidence = 99
?: population standard deviation = 1000
('z critical value') from Look-up Table for 99% = 2.58
The Look-up in the Table for the Standard Normal Distribution utilizes the Table's cummulative 'area' feature. The Table shows positve and negative values of ('z critical') but since the Standard Normal Distribution is symmetric, only the magnitude of ('z critical') is important.
For a Level of Confidence = 99% the corresponding LEFT 'area' = 0.5. And due to Table's symmetric nature, the corresponding RIGHT 'area' = 0.5 The ('z critical') value Look-up is 2.58
significant digits = 2
Margin of Error = ('z critical value') * ?/SQRT(n)
n = Sample Size
Algebraic solution for n:
n = [('z critical value') * ?/Margin of Error]
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