Question

The Central Limit Theorem Stock Prices The following table lists initial public offering (IPO) stock prices for all 1999 stocks that at least doubled in value during the first day of trading. This is historical data.

$17.00 $23.00 $14.00 $16.00 $12.00 $26.00 $20.00 $22.00 $14.00 $15.00 $22.00 $18.00 $18.00 $21.00 $21.00 $19.00 $15.00 $21.00 $18.00 $17.00 $15.00 $25.00 $14.00 $30.00 $16.00 $10.00 $20.00 $12.00 $16.00 $17.44 $16.00 $14.00 $15.00 $20.00 $20.00 $16.00 $17.00 $16.00 $15.00 $15.00 $19.00 $48.00 $16.00 $18.00 $9.00 $18.00 $18.00 $20.00 $8.00 $20.00 $17.00 $14.00 $11.00 $16.00 $19.00 $15.00 $21.00 $12.00 $8.00 $16.00 $13.00 $14.00 $15.00 $14.00 $13.41 $28.00 $21.00 $17.00 $28.00 $17.00 $19.00 $16.00 $17.00 $19.00 $18.00 $17.00 $15.00 $14.00 $21.00 $12.00 $18.00 $24.00 $15.00 $23.00 $14.00 $16.00 $12.00 $24.00 $20.00 $14.00 $14.00 $15.00 $14.00 $19.00 $16.00 $38.00 $20.00 $24.00 $16.00 $8.00 $18.00 $17.00 $16.00 $15.00 $7.00 $19.00 $12.00 $8.00 $23.00 $12.00 $18.00 $20.00 $21.00 $34.00 $16.00 $26.00 $14.00

Using this data:

a. In words, X=

b. Find the following:

i. μX =

ii. σX =

iii. n =

c. Construct a histogram of the distribution. Start at x=−0.50. Make bar widths of $5.

d. In words, describe the distribution of stock prices.

e. Randomly average 5 stock prices together. (Use a random number generator.) Continue averaging 5 pieces together until you have 15 averages. List those 15 averages.

f. Use the 15 averages from (e) to calculate the following:

i. mean of x (x−) =

ii. Standard deviation of mean of x (sx−−)=

g. Construct a histogram of the distribution of the averages. Start at x=−0.50. Make bar widths of $5.

h. Does this histogram look like the graph in (c)? Explain any differences.

i. In 1 - 2 complete sentences, explain why the graphs either look the same or look different?

j. Based upon the theory of the Central Limit Theorem, X−−~

Answer 1

Solution:-) We have used R for the following problem

On L.H.S. is the R-Output and R.H.S. is the R-Code

So,

a. In words, X=  Initial public offering (IPO) stock prices for all 1999 stocks

b. The following are

i. ?X = 17.60556

ii. ?X = 5.707057

iii. n = 117

c) The histogram is given above (Note that the rabge of X is -0.5 to 50 but as between their are no values, So the range is not visible in R from -0.5 )

d) You can see from the above histogram that the distribution of X is somewhat symmeterical , it tends to normal distribution.

e. ) We have Randomly average 5 stock prices together. (Using sample command .) Continue averaging 5 pieces together until you have 15 averages. those 15 averages are

20.800 18.600 20.200 21.400 18.400 19.400 17.800 17.400 17.000 19.688 17.000 18.800 18.400 17.600 15.488

f. The following:

i. mean of x (x?) =17.02

ii. Standard deviation of mean of x (sx??)=1.592602

g. You can see the histogram above

h. There are differences in the range between sample and population as the sample range is mainly from 10-30, but the shape of distribuition is somewhat similar,So , we can see that sample show some characteristics of population.

i. The graphs look either the same , due to central limit theorm because as the sample size increase the distribution tends to normal distribution as we have averaged 5 values then take sample up to 15. So, the sample explain most of the characteristics of population.

j. Based upon the theory of the Central Limit Theorem, X??~N(17.60,25) (Normal distribution)