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1 Confidence intervals

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2 Point and interval estimators There are two kind of estimators: Point Interval Point estimator: single statistics used for rstiamte a paramether of the population. Sample mean is a point estimator for the mean of the population , sample variance is a point estimator for the variance of the population 2, ecc.

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3 Point and interval estimators Interval estimator: interval of values that has a certain probability or confidence to contain the true value of the paramenter of the population. The level of confidence is usually (1- )% where is the probability that the value stays in the tails of the distribution, outside the confidence interval.

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4 Intervallo di confidenza per la media noto il valore dello scarto quadratico medio La statistica per costruire intervalli di confidenza per la media è Ovvero una distribuzione Normale standardizzata, “indipendentemente” dalla distribuzione originale della variabile X (campioni sufficientemente grandi). Da tale distribuzione scaturiscono gli estremi dell’intervallo di confidenza per la media.

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5 Confidence interval for the mean known σ of the poplulation When the population is normally distributed, the distribution of the mean is also normal When population variable X is normally distributed and is known, a 100(1 − α)% confidence interval for μ is given by

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6 Confidence interval Normal curve for Z with a level of confidence of 95% Normal curve for Z with a level of confidence of 99%

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Example Bolts produced from a firm have an unknown mean diameter, its variance is 0.01. We keep a sample of n=1000 bolts and we observe a mean diameter of 1.2 cm. Find the confidence interval at 99% with a fixed level of confidence of 99%. 1-α=0.99→ α=0.01 → α/2=0.005 →1-α/2=0.995 From table of Normal distribution Z(0.995) =2.576 So the interval is

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8 Confidence interval for the mean unknown σ of the poplulation Given that σ is not know we need to use its estiamator s. If we consider the ratio then the random variable t has the Student’s t distribution with n − 1 degrees of freedom. A 100(1−α )% confidence interval for μ is given by where t α/2 is the upper α/2 point of the Student’s t distribution with n − 1 degrees of freedom.

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9 If n is very large the t distribution is very close to a Normal distribution. Confidence interval for the mean unknown σ of the poplulation

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10 The table of the Student’s t distribution give the probability (area) on the right of the indicated value. Confidence interval for the mean unknown σ of the poplulation

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