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# A random sample will be drawn from a normal distribution,

ISBN: 9780073401331 38

## Solution for problem 12E Chapter 4.12

Statistics for Engineers and Scientists | 4th Edition

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Statistics for Engineers and Scientists | 4th Edition

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Problem 12E

A random sample will be drawn from a normal distribution, for the purpose of estimating the population mean µ. Since µ is the median as well as the mean, it seems that both the sample median m and the sample mean  are reasonable estimation. This exercise is designed to determine which of these estimators has the smaller uncertainty.

a. Generate a large number (at least 1000) samples of size 5 from a N(0, 1) distribution.

b. Compute the sample medians  for the 1000 samples.

c. Compute the mean  and the standard deviation

d. Compute the sample means for the 1000 samples.

e. Compute the mean and standard deviation of

f. The true value of µ is 0. Estimate the bias and uncertainty (σm) in m. (Note: In fact, the median is unbiased, so your bias estimate should be close to 0.)

g. Estimate the bias and uncertainty in. Is your bias estimate close to 0? Explain why it should be. Is your uncertainty estimate close to 1/? Explain why it should be.

Step-by-Step Solution:

Step 1 of 4:

A random sample drawn from a normal normal distribution, for the purpose of estimating mean  , here mean and median is same. So both estimate of sample median m and and sample mean  should be reasonable. An exercise is conducted to find which of this estimate has the smaller uncertainty.

Step 2 of 4:

We have to generate a sample 1000 of size 5 from N(0,1)distribution.

We can generate a sample 1000 of size 5 from N(0,1) by using minitab as

Calc- Random data- Distribution- parameters( with sample size) -Ok.

Generated samples will be different from ours.

(b)  We have to compute the sample medians  m1, m2, …,m1000 for1000 samples.

We can find the sample medians for 1000 samples as:

Calc- Calculator- expression (Median of rows(specify the rows)) - Ok.

(c) We have to find the mean and standard deviation of the estimated medians of 1000 samples

We can find the mean and standard deviation of the estimated medians as

Stat - Basic Statistics- Descriptive Statistics- variable -statistics(mean, standard deviation) - Ok.

mean  -0.0129

Standard deviation 0.5540.

Therefore the mean and standard deviation  of the  estimated medians are  -0.0129, .

Step 3 of 4

Step 4 of 4

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A random sample will be drawn from a normal distribution,

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