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QUESTION

# Generate and store in column c1 4,000 values from the exponential distribution with scale parameter=6 by typing the following (in the command...

Generate and store in column c1 4,000 values from the exponential distribution with scale parameter=6 by typing the following (in the command session):

random 4000 c1;

exponential 6.

Note: The mean µ and the standard deviation σ of such a distribution are both equal to the scale parameter and this is the value you are asked to enter in the command above.

[3] a. Use desc command to find the sample mean ????̅=—————,

the median———, and the sample standard deviation s =————— for these 4,000 data.

[3] b. What value(s) do ????̅ and s seem to be close to?———————-

Why so?-------------------------------------------------------------------------------------------------------------------------------

[3] c. Generate a histogram for the 4,000 values you generated from this exponential distribution. What is the shape of the distribution?---------------------------------------------------------------------- Is this confirmed by the values of the mean and the median you found in (a.)?

————Explain——————————————-.

2.Normal distribution: Generate and store in column c2 5,000 values from the normal distribution with mean 5 and standard deviation 3.5 as follows:

random 5000 c2

normal 5, 3.5

[3] a. Generate the histogram for these data. What is the shape of this histogram? ---------------------------------------------------------------

[3] b. What is the value on the horizontal axis around which the histogram seems to be symmetric? X=---

[3] c. Use Minitab to find the sample mean ????̅=————— and the standard deviation s--------------------- for the data you generated.

[3] d. What values are ????̅and s respectively close to?---------------- and ----------------. Why so?------------------ ---------------------------------------------------------

3. Central limit theorem (CLT) at work (You can open a new Minitab worksheet, simply by typing new). Generate and store in columns c3-c1002 120 horizontal samples, each of size n = 1000, from exponential distribution with mean µ = 8 as follows:

random 120 c3-c1002;

expo 8.

Note This may (or may not!) take a few moments as you are generating 1000×120=120,000 values! Create and store in column c1 the 120 values of ????̅based on the 120 horizontal samples, each of the same size n = 1000 as follows:

rmean c3-c1002 c1

[2]a. Generate the boxplot of the first sample c3. According to the median position and/or the outliers, what can you conclude about the shape of this data set?-----------------------------------

[3]b. Use desc command to find the sample mean---------and the median-------of c3. Do they confirm your diagnostic in part (a) for the shape above?---------------Explain.-------------------------------------------------

[2]c. Generate the histogram for the data in column c1. What can you conclude about the shape of data in c1?————————-

[3]d. Use desc command to find sample mean-----------and sample median----------- of c1.

Do they confirm your findings in (c.) about the shape of data in c1?---------------

[3] e. What is the sample standard deviation of c1?------------------------------------ What values do the sample mean and the sample standard deviation of c1 seem to be respectively close to?---------------------- and --------------------------. Why so?---------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------

4.Confidence interval (CI) for a population mean: We want to build 150 confidence intervals (CIs) with confidence level (1 − α)100% = 95% for the mean µ of a Poisson distribution via the following steps:

Step 1. Open a new worksheet. Generate and store in columns c6-c505 150 samples of size 500 each, from Poisson with parameter µ = 6.5 as follows:

random 150 c6-c505;

poisson 6.5.

Step 2. Use columns c4 and c5 to store respectively the means and the standard deviations of the 150 horizontal samples you generated in step 1, as follows:

rmean c6-c505 c4

rstd c6-c505 c5

Step 3. Store the lower bound and the upper bound of each of your 95% CIs in c2 and c3 respectively by typing successively:

let c2=c4-1.96*c5/sqrt(500)

let c3=c4+1.96*c5/sqrt(500)

Step 4. Then create a column c1 containing 1 or 0 according to whether or not the corresponding interval [c2 ,c3] covers µ or not, by typing in the following logical function:

let c1=(c2 <6.5 and c3 >6.5)

Finally sum up the entries of column c1 to find out how many (and what percent of ) CIs did cover the value µ = 6.5 by typing: tally c1;

counts;

percent.

[3] a. How many confidence intervals (and out of what) did contain the true value µ = 6.5?----------------- and What is the percent of intervals that did cover µ = 6.5?------------------

[3] b. What value should the percent of intervals you found in (a) be close to?---------------Explain.---------- ------------------------------------------------------------------------------------ Was this the case?----------------Explain-------------------------------------------------------------------------------