Question

i.Bias of Sample Mean Draw 20 samples from the normal distribution N(5, 4). Compute the mean of your 20 samples. Report the bias of the sample mean

ii. Variance of Sample Mean (Continue of problem i) To estimate the variance of the sample mean, we need to draw many different samples of size 20. Now, we draw 1000 times a sample of size 20. Store all the 1000 sample means. Report the variance of the estimated sample mean. Hint: To draw the samples 1000 times, we need to use ”for loop” in R. Firstly we create a vector of length 1000. Then we store each sample mean into the vector. Then you can estimate the variance of the sample mean with the variance function in R. sample.mean = rep(0, 1000) for(i in 1:1000) { s1 = rnorm(20, mean, sd) sample.mean[i] = mean(s1)

iii.

Answer #1

(4)(b) Using *set.seed(124)* and the Nile data, generate 1000
random samples of size n = 16, with replacement. For each sample
drawn, calculate and store the sample mean. This can be done with a
for-loop and use of the *sample()* function. Label the resulting
1000 mean values as "sample1". **Repeat these steps using
*set.seed(127)* - a different "seed" - and samples of size n =
64.** Label these 1000 mean values as "sample2". Compute
and present the means, sample standard...

Use R.
Generate a random sample with n=15 random observations from an
exponential distribution with mean=1.
Calculate the sample median, which is an estimator of the
population median.
Use bootstrap (nonparametric, with B=1000) methods to estimate
the variance of the estimator for the population median.
use the Monte Carlo method, e.g. generate 1000 samples of size
15 to estimate the true variance of the median estimator. Compare
and comment on your results.

if we draw samples of sizes 10 many times and form a
distribution of sample mean, state the distribution of the sample
mean and provide the reason for your conclusion; calculate the mean
of the sample means and its standard deviation.
-0.021 0.029 -0.009 -0.002 0.002 -0.006 0.006 -0.064 0.023
0.031

Question: Eleven:
(a) In a sample of 16 observations from a normal distribution
with mean 150 and variance of 256. Find the following
probabilities
i. Ρ ( Χ ≤160) ii. Ρ ( Χ ≥142)
(b) The age of employees in a company follows a normal
distribution with its mean and variance as 40 years and 121 years,
respectively. If a random sample of 36 employees is taken from a
normal population of size 1000, what is the probability that the...

Data will be drawn independently from a normal distribution N(5,
4), i.e. with mean 5 and variance 4. A single observation will be
termed X. The sample mean (average) of n observations will be
termed X. If the sample size drawn is n = 100, derive the
distribution of X.

Using R, generate 10 different samples of size 20 from a
standard normal distribution and generate qq plots for each sample.
Include two qq plots for your answer; one that shows the sample
that is most and one that shows the sample that is least similar to
what we would expect from a normal population. Make sure to label
which is which and explain what characteristics of the graphs you
used to choose which was which.

The weak law of large numbers states that the mean of a sample
is a consistent estimator of the mean of the population. That is,
as we increase the sample size, the mean of the sample converges in
probability to the expected value of the distribution that the data
comes from, provided that expected value is finite. Consider a
numerical example, a Student t distribution with n = 5, the same as
we have already seen earlier in this module....

Samples of n =5 units are taken from a process every hour. The
x̄ and R̄ values for a particular quality characteristic are
determined. After 25 samples have been collected, we calculate x̄ =
20 and R̄ = 4.56.
(a) What are the three- sigma control limit for x̄ and R?
(b) Both charts exhibit control. Estimate the process standard
deviation.
(c) Assume that the process output is normally distributed. If
the specifications are 19 ± 5, what are your...

Make in Excel three tables size 50 times 50, 250 times 50, 1000
times 50 with Bernoulli distributed random numbers for p(1) = 1/3.
Use the generated tables for calculating new column as t he average
of the first 50 columns. Random numbers in this column will have
distribution close to the Normal distribution. Find population
probability distribution functions, means, variances and standard
deviations and compare its with sample frequency distribution
functions, averages, sample estimation of variances and standard
deviations...

Conducting a Simulation
For example, say we want to simulate the probability of getting
“heads” exactly 4 times in 10 flips of a fair coin.
One way to generate a flip of the coin is to create a vector in
R with all of the possible outcomes and then randomly select one of
those outcomes. The sample function takes a vector of elements (in
this case heads or tails) and chooses a random sample of size
elements.
coin <- c("heads","tails")...

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