Statistics Project

My project instructions

One day at work, Colleen and Gail start talking about the stock market and how volatile it has
been lately. Colleen asks Gail, “The market has been acting very chaotically. What do you think
is going on?” Gail replies, “Who knows. These are all short-term movements and therefore they
are completely random. On the business channels, experts try to explain every single up and
down. One expert states a reason for the market moving up and the next day another expert
uses the same reason to explain why the market is going down.”
Colleen then asks, “Well, how do you invest in a market like this then? I have some extra
savings that I want to put into action. I feel if we apply some statistics to this, we can come up
with some simple techniques that could work.” Gail responds, “There are thousands of stocks
and methods to choose from. Big institutions, private investors, and companies spend billions
of dollars to do research on how they should invest their assets. Do you want to compete with
them?”
Colleen responds, “Absolutely not. But you said it yourself that with all their expertise they still
cannot tell the future! No one can. So, let’s try to come up with a couple of methods for fun and
see how we can predict a range for the near-term market value. In fact, let’s concentrate on the
S&P 500 index which is a good indicator for the stock market.”
Gail shakes her head, “Let’s give it a try. To keep the values smaller why don’t we concentrate
on an ETF (Exchange Traded Fund) that is based on the S&P 500. The common one that
everyone uses is SPY, it mimics the S&P 500 movement approximately.” Colleen says, “That
sounds good. Let’s look at the daily chart of SPY and see what we can come up with.” They
quickly get behind a computer terminal and check out the prices of SPY for the last three
months. Looking at a bar graph, Colleen asks, “Which of these values should we take as the
price: the open, the close, the low or the high?”
Gail responds, “I assume that we are only going to look at the price action. It looks like the
trading volume is large enough for each day that perhaps we can ignore it for now. Back to your
question, the highs and lows are really the daily extremes. Let’s find a 90% confidence interval
for the highs and one for the lows. Then we can use the upper value of the confidence interval
for the highs as a sell signal and the lower value of the confidence interval for the lows as a buy
signal. There you go, you have one method.”
Colleen thinks a little and says, “That’s a good one, but it is too conservative. I want to get into
the daily action more often. You agree that the opening and closing prices are more
representative of the ‘actual’ value of SPY, right?” Gail replies, “That’s right.” Colleen continues,
“Why don’t I take the midpoint for opening and closing prices of each day and then do a 99%
confidence interval on the midpoints of the opening and closing prices. Once I have that range,
then I can sell SPY when its price reaches the upper limit of the confidence interval and buy SPY
when it comes close to the lower limit. That works, doesn’t it?”
Gail responds, “On paper, everything works. We really need to do some tests before you even
think about applying either one of these methods. There are many other things that we should
consider too.” Colleen looks at the clock, “I know. Don’t worry, I won’t be doing any investing
right away. I will do some more research. Lunch time is up, we need to go.” As they depart for
work, they both wonder how well their methods would work.
Determine a 90% confidence interval for the average daily low prices of SPY and a 90%
confidence interval for the average daily high prices. Find the midpoint of the opening and
closing values for those same days and call it the “Price” of SPY for that day. Compute a 99%
confidence interval for the average “Price” of SPY. (Make sure to get the data from a reliable
financial site. The last 30-trading day data on SPY will suffice.) Describe at what values SPY
should be bought or sold according to Gail’s method. How about the trading values for
Colleen’s method?

Solve and write a detailed solution