Answered: 400 500 600 700 800 900 1,000 1,100…

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

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Question

To estimate a population mean μ, the sample mean x¯ is often used as an estimator. However, a different estimator is called the sample midrange, given by the formula sample minimum + sample maximum/ 2.

(a) The following table shows the values, in thousands of dollars, of 40 randomly selected houses in the city.

(i) Calculate the sample midrange for the data.

(ii) Explain why the sample midrange might be preferred to the sample mean as an estimator of the population mean.

(b) To investigate the sampling distribution of the sample midrange, a simulation is performed in which 100 random samples of size n=40 were selected from the population of house values. For each sample, the sample midrange was calculated and recorded on the following dotplot. The mean of the distribution of sample midranges is $617,000 with standard deviation $136,000. Based on the results of the simulation, explain why the sample mean might be preferred to the sample midrange as an estimator of the population mean.

400
500
600
700
800
900
1,000
1,100
Sample Midrange (thousands of dollars)

34
38
95
137
140
155
155
207
110
169
209
217
314
314
323
343
347
349
363
274
369
388
389
397
416
448
483
487
373
450
488
516
571
595
600
738
762
769
863
1,084

Definition Definition Measure of central tendency that is the average of a given data set. The mean value is evaluated as the quotient of the sum of all observations by the sample size. The mean, in contrast to a median, is affected by extreme values. Very large or very small values can distract the mean from the center of the data. Arithmetic mean: The most common type of mean is the arithmetic mean. It is evaluated using the formula: μ = 1 N ∑ i = 1 N x i Other types of means are the geometric mean, logarithmic mean, and harmonic mean. Geometric mean: The nth root of the product of n observations from a data set is defined as the geometric mean of the set: G = x 1 x 2 ... x n n Logarithmic mean: The difference of the natural logarithms of the two numbers, divided by the difference between the numbers is the logarithmic mean of the two numbers. The logarithmic mean is used particularly in heat transfer and mass transfer. ln x 2 − ln x 1 x 2 − x 1 Harmonic mean: The inverse of the arithmetic mean of the inverses of all the numbers in a data set is the harmonic mean of the data. 1 1 x 1 + 1 x 2 + ...

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