Using the Central Limit Theorem · Exercise 26

The closing stock prices of 35 U.S. semiconductor manufacturers are given as follows:

$$\begin{array}{cc}\text{Company}& \text{Closing Stock Prices}\\ \text{1}& \text{8.625}\\ \text{2}& \text{30.25}\\ \text{3}& \text{27.625}\\ \text{4}& \text{46.75}\\ \text{5}& \text{32.875}\\ \text{6}& \text{18.25}\\ \text{7}& \text{5}\\ \text{8}& \text{0.125}\\ \text{9}& \text{2.9375}\\ \text{10}& \text{6.875}\\ \text{11}& \text{28.25}\\ \text{12}& \text{24.25}\\ \text{13}& \text{21}\\ \text{14}& \text{1.5}\\ \text{15}& \text{30.25}\\ \text{16}& \text{71}\\ \text{17}& \text{43.5}\\ \text{18}& \text{49.25}\\ \text{19}& \text{2.5625}\\ \text{20}& \text{31}\\ \text{21}& \text{16.5}\\ \text{22}& \text{9.5}\\ \text{23}& \text{18.5}\\ \text{24}& \text{18}\\ \text{25}& \text{9}\\ \text{26}& \text{10.5}\\ \text{27}& \text{16.625}\\ \text{28}& \text{1.25}\\ \text{29}& \text{18}\\ \text{30}& \text{12.87}\\ \text{31}& \text{7}\\ \text{32}& \text{12.875}\\ \text{33}& \text{2.875}\\ \text{34}& \text{60.25}\\ \text{35}& \text{29.25}\end{array}$$

(a) In words, Χ = ______________.
(b) (a) $\bar{x}$ = _____
(b) s_x = _____
(c) n = _____
(c) Construct a histogram of the distribution of the averages. Start at x = –0.0005. Use bar widths of 10.
(d) In words, describe the distribution of the stock prices.
(e) Randomly average five stock prices together. (Use a random number generator.) Continue averaging five prices together until you have 10 averages. List those 10 averages.
(f) Use the 10 averages from part (e) to calculate the following:
(a) $\bar{x}$ = _____
(b) s_x = _____
(g) Construct a histogram of the distribution of the averages. Start at x = –0.0005. Use bar widths of 10.
(h) Does this histogram look like the graph in Part (c)?
(i) In one or two complete sentences, explain why the graphs either look the same or look different.
(j) Based on the theory of the central limit theorem, $\bar{X}$ ~ _____(_____, ____).

This is an open-ended exercise: its source gives no single answer.

OpenStax, Statistics · §7.3 Using the Central Limit Theorem · Exercise 26