Calculating the Confidence interval for a mean using a formula

How to calculate the confidence interval

Please note thatinterval(l1 or l2), the central area of ​​which will be the estimate of l *, and also in which the true value of the parameter is with probability alpha, will be just confidenceintervalohm or the corresponding confidence value alpha. In this case, l * itself will be related to point estimates. For example, according to the results of any sample values ​​of the random value X {x1, x2, ..., xn}, it is necessary to calculate the unknown parameter of the indicator l, on which the distribution will depend. In this case, obtaining an estimate of a given parameter l * will consist in the fact that for each sample it will be necessary to put a certain value of the parameter in accordance, that is, to create a function of the observation results of the indicator Q, the value of which will be taken equal to the estimated value of the parameter l * in the form of a formula : l * = Q * (x1, x2, ..., xn).
Please note that any function according to the observation results is called statistics.At the same time, if it fully describes the parameter (phenomenon) under consideration, then it is called sufficient statistics. And since the results of observations are random, l * will also be a random variable. The task of calculating statistics should be made taking into account the criteria of its quality. Here it is necessary to take into account that the law of distribution of the estimate is quite definite, if the probability density distribution W (x, l) is known.
You can calculate the trustintervalsimple enough if you know the law on the distribution of estimates. For example, trustintervalestimates regarding the expectation (average value of a random value) mx * = (1 / n) * (x1 + x2 +… + xn). This estimate will be unbiased, that is, the expectation or average value of the indicator will be equal to the true value of the parameter (M {mx *} = mx).
You can establish that the variance of the estimate for the expected value: bx * ^ 2 = Dx / n. On the basis of the limit central theorem, we can conclude that the distribution law for this estimate is Gaussian (normal). Therefore, for calculations you can use the indicator Ф (z) - the integral of probabilitiesIn this case, select the length of the trustintervaland 2ld, so you get: alpha = P {mx-ld (using the property of the probability integral by the formula: Ф (-z) = 1 - Ф (z)).
Build a trustintervalestimates of expectation: - find the value of the formula (alpha + 1) / 2; - select a value equal to ld / sqrt (Dx / n) from the probability integral table; - take the estimate of the true variance: Dx * = (1 / n) * ( (x1 - mx *) ^ 2+ (x2 - mx *) ^ 2 + ... + (xn - mx *) ^ 2); - determine ld; - find the trustintervalaccording to the formula: (mx * -ld, mx * + ld).

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