Confidence Intervals

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Confidence Intervals

Consider the following question: someone takes a sample from a population and finds both the sample mean and the sample standard deviation. What can he learn from this sample mean about the population mean?
This is an important problem and is addressed by the Central Limit Theorem. For now, let us not bother about what this theorem states but we will look at how it could help us in answering our question.

The Central Limit Theorem tells us that if we take very many samples the means of all these samples will lie in an interval around the population mean. Some sample means will be larger than the population mean, some will be smaller. The Central Limit Theorem goes on to state that 95% of the sample means
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A brief note: Statistical software often gives slightly different but more accurate values. Let us stick with the consensus of the majority of people working with elementary statistics and use the critical values given above. Summarized:

|Confidence Interval |Critical Value |
|90% |-1.65 |
|95% |-1.96 |
|99% |-2.58 |

Due to the perfect symmetry of the curve we will find the same but this time positive z-values for the right tail. [pic]

Now we can start calculating confidence intervals.

Sample Mean and Standard Deviation are known, in the case of Sample Size [pic]

The confidence interval for the population mean μ in this situation is given by [pic].

Example 1:
We are looking at a very large group of tests. Forty five tests were selected from this group. The mean of the sampled tests is 58 and the standard deviation is 13. Find the 95% confidence interval for the mean grade for the whole class.

[pic]. This means that we are 95% confident that the population mean for all these tests is between [pic] and [pic] or more simply said, between 54.2 and 61.8.

The calculator keystrokes on the TI 30XIIB for that part of the calculation that follows the ( in example 1 are:

1.96 x 13


Please note that taking a sample gives us information

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