Which of the following is an estimate of the standard error of the mean?Group of answer choicesThe standard deviation of returns divided by the square root of the number of observationsThe average annual rate of return divided by the square root of the number of observationsThe variance of returns divided by the square root of the number of observationsThe variance divided by the number of observations

29.The standard deviation of the sample means is equal to which of the following?  A. population standard deviation divided by the square root of n (sample size)  B. sample standard deviation divided by n (sample size)  C. sample standard deviation divided by the square root of n (sample size)  D. population standard deviation divided by n (sample size)

Standard Error is used primarily in which of the following?Question 1Answera.Estimating the population mean from the sample mean.b.Determining the range of sample means.c.Calculating the variance of a sample.d.Measuring the effect size of an experiment.

The standard deviation of a sampled population is also called the standard error of the sample mean.Group of answer choicesTrueFalse

s the formula for the mean square error?Review Later

Let us first define the standard deviation s. Suppose weperform N measurements 1, 2, · · · , N with the average¯. Then the deviation of each measurement is given byδ =  − ¯ with  = 1, 2, · · · , N. The standard deviation s iss =√√√ ∑N=1(δ)2N − 1When we report the average value of N measurements,the uncertainty we should associate with this averagevalue is the standard error.σ = spN =√√√√∑N=1(δ)2N(N − 1)The standard error is smaller than the standard deviationby a factor of 1/pN, since the statistical uncertainty canbe reduced by large number of measurements. Also it isuseful to write σ2 = δ2 ≡ 1N(N − 1)N∑=1δ2 .Suppose we want to determine a quantity  = ƒ (, ),which depends on  and . We want to know the error in = ƒ (, ) if we measure  and  with errors σ and σ .Using the Taylor expansion, we can obtain the law of theerror propagation as follows(δ)2 = ∂ƒ∂2(δ)2 + ∂ƒ∂2(δ)2 + 2 ∂ƒ∂∂ƒ∂(δδ)If the measurements of  and  are uncorrelated, then,on the average, we should expect to find equal distribu-tions of positive and negative values for this term, and weshould expect (δδ) = 0. At the end of the day, using thedefinition of the standard error σ, we can obtainσ =√√√ ∂ƒ∂2σ2 + ∂ƒ∂2σ2Exercise problems: Now find the standard error σ in = ƒ (, ) as a function of the errors in σ and σ for thefollowing functions:(a)  =  +  (0.5 pts)You can find the absolute uncertainty of the sum(or difference) is the root square sum of theindividual absolute uncertainties when adding (orsubtracting).(b)  =  ×  (0.5 pts)(c)  = /  (1 pt)You can find that the relative uncertainty of the prod-uct (quotient) is the root square sum of the individualrelative uncertainties.(d)  = 2 (1 pt)(e)  =  exp(c) with c constant. (0.5 pts)(f)  = 1/ 