To find:
A 99% confidence interval for the mean
Answer to Problem 14E
Solution:
A 99% confidence interval for the mean face amount of an individual life insurance policy is in the interval,
Explanation of Solution
Approach:
Given a sample with standard deviation
The true population parameters probability lies in this interval range which is known as the confidence level
At a certain level of confidence, the interval in which the maximum error can be observed is known as the confidence interval.
Estimating parameters of a large population or a population following normal distribution is done using the Student’s
The
The degrees of freedom is the attribute used to obtain the appropriate curve for the
The value of
Hence this is a case of two-tailed t-distribution.
The maximum distance from point estimate that the confidence interval covers is margin of error and is given by:
Given the sample mean
Calculation:
The sample means is given by:
Substituting the values of life insurance policies in the formula above
150, 000 | 171758.62 | -21, 758.62 | 473437544.3 |
150, 000 | 171758.62 | -21, 758.62 | 473437544.3 |
150, 000 | 171758.62 | -21, 758.62 | 473437544.3 |
150, 000 | 171758.62 | -21, 758.62 | 473437544.3 |
150, 000 | 171758.62 | -21, 758.62 | 473437544.3 |
150, 000 | 171758.62 | -21, 758.62 | 473437544.3 |
151, 000 | 171758.62 | -20, 758.62 | 430920304.3 |
152, 000 | 171758.62 | -19, 758.62 | 390403064.3 |
152, 000 | 171758.62 | -19, 758.62 | 390403064.3 |
153, 000 | 171758.62 | -18, 758.62 | 351885824.3 |
153, 000 | 171758.62 | -18, 758.62 | 351885824.3 |
154, 000 | 171758.62 | -17, 758.62 | 315368584.3 |
155, 000 | 171758.62 | -16, 758.62 | 280851344.3 |
158, 000 | 171758.62 | -13, 758.62 | 189299624.3 |
158, 000 | 171758.62 | -12, 758.62 | 162782384.3 |
158, 000 | 171758.62 | -12, 758.62 | 162782384.3 |
160, 000 | 171758.62 | -11, 758.62 | 138265144.3 |
160, 000 | 171758.62 | -11, 758.62 | 138265144.3 |
160, 000 | 171758.62 | -11, 758.62 | 138265144.3 |
162, 000 | 171758.62 | -9, 758.62 | 95230664.3 |
163, 000 | 171758.62 | -8, 758.62 | 76713424.3 |
163, 000 | 171758.62 | -8, 758.62 | 76713424.3 |
163, 000 | 171758.62 | -8, 758.62 | 76713424.3 |
165, 000 | 171758.62 | -6, 758.62 | 45678944.3 |
165, 000 | 171758.62 | -6, 758.62 | 45678944.3 |
168, 000 | 171758.62 | -3, 758.62 | 14127224.3 |
168, 000 | 171758.62 | -3, 758.62 | 14127224.3 |
168, 000 | 171758.62 | -3, 758.62 | 14127224.3 |
171, 000 | 171758.62 | -758.62 | 575504.3044 |
171, 000 | 171758.62 | -758.62 | 575504.3044 |
172, 000 | 171758.62 | 241.38 | 58264.3044 |
172, 000 | 171758.62 | 241.38 | 58264.3044 |
172, 000 | 171758.62 | 241.38 | 58264.3044 |
173, 000 | 171758.62 | 1, 241.38 | 1541024.304 |
174, 000 | 171758.62 | 2, 241.38 | 5023784.304 |
175, 000 | 171758.62 | 3, 241.38 | 10506544.3 |
175, 000 | 171758.62 | 3, 241.38 | 10506544.3 |
175, 000 | 171758.62 | 3, 241.38 | 10506544.3 |
176, 000 | 171758.62 | 4, 241.38 | 17989304.3 |
182, 000 | 171758.62 | 10, 241.38 | 104885864.3 |
182, 000 | 171758.62 | 10, 241.38 | 104885864.3 |
183, 000 | 171758.62 | 11, 241.38 | 126368624.3 |
185, 000 | 171758.62 | 13, 241.38 | 175334144.3 |
185, 000 | 171758.62 | 13, 241.38 | 175334144.3 |
190, 000 | 171758.62 | 18, 241.38 | 332747944.3 |
190, 000 | 171758.62 | 18, 241.38 | 332747944.3 |
195, 000 | 171758.62 | 23, 241.38 | 540161744.3 |
195, 000 | 171758.62 | 23, 241.38 | 540161744.3 |
195, 000 | 171758.62 | 23, 241.38 | 540161744.3 |
196, 000 | 171758.62 | 24, 241.38 | 587644504.3 |
200, 000 | 171758.62 | 28, 241.38 | 797575544.3 |
200, 000 | 171758.62 | 28, 241.38 | 797575544.3 |
200, 000 | 171758.62 | 28, 241.38 | 797575544.3 |
200, 000 | 171758.62 | 28, 241.38 | 797575544.3 |
200, 000 | 171758.62 | 28, 241.38 | 797575544.3 |
202, 000 | 171758.62 | 30, 241.38 | 914541064.3 |
202, 000 | 171758.62 | 30, 241.38 | 914541064.3 |
163, 000 | 171758.62 | -8, 758.62 | 76713424.3 |
Form the above table we get,
Now the standard division is,
Level of confidence is 99%
Where
Critical t value at
Now substitute these values in margin of error.
Then, the margin of error is calculated as:
Then, interval is:
Conclusion:
A 95% confidence interval for the mean fastball pitching speed of all high school is in the interval
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Chapter 8 Solutions
Beginning Statistics, 2nd Edition
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