Statistical laws, normal law, binomial law, confidence intervals, central limit theorem, statistical inference
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Abstract
This document explores the application of statistical laws, including the normal and binomial laws, to calculate confidence intervals and make inferences about population parameters. It covers exercises on approximating the normal law, binomial confidence intervals, and the central limit theorem.
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[...] So = follows a normal law of mean and standard deviation . We have an empirical realization of the mean: = 2.53 cm 2. a. We are looking for a and b such that P(a ≤ ≤ = 0.95. After centering and reduction we know that follows a normal law with a mean of zero and a standard deviation of 1. By the symmetry of the normal law and the properties of the quantile function, we find a = et b = with u the 1st quantile 0.975 of the normal law. [...]
[...] We use exercise 3 as the reasoning is the same and we find a confidence interval ; at 95% with: a = 0.0026 - 1.96 * = 0.0017 and b = 0.0026 + 1.96 * = 0.0035 Here is the result in grams per liter, we go back to milligrams per liter and we find 1.7 and 3.5 milligrams per liter 2. We use here the exercise we assume that the concentration follows a normal law with a mean of 2.6 and a standard deviation of 0.3 and we find therefore a confidence interval ; at 95% with: a = 2.6 - 1.96 * = 2.43 and b = 2.6 + 1.96 * = 2.77 3. The amplitude is weaker so the result is much more precise Exercise No. 5 As previously, it is sufficient to build a confidence interval. [...]
[...] This quantile is approximately 1.96. On has a = 2.53 - 1.96 * = 2.518 and b = 2.53 +1.96 * = 2.542 b. Its amplitude is b - a = 0.025 3. a. We are now looking for a and b such that P(a ≤ ≤ = 0.98. By the same reasoning, we find a = et b = with u the 1st quantile 0.99 of the normal law. This quantile is approximately 2.33. On has a = 2.53 - 2.33 * = 2.515 and b = 2.53 +2.33* = 2.545 b. [...]
[...] We can therefore apply the central limit theorem: noting T the average number of calls per minute, calculated from the 24 sequences. T can be approximated by a normal law of expectation and standard deviation 3. We make the same reasoning as in exercise and we therefore find a confidence interval ; at the 95% level with: a = 6 - 1.96 * = 5.02 and b = 6 + 1.96 * = 6.98 Exercise n°3 We will use the binomial law here. [...]
