This post starts teaches how to conduct a hypothesis test about proportions with data on stock declines for two funds.
Stock Decline Proportions
Question: The table below has information on the
number of times a largecap and a smallcap fund declines in a 10day
period. The data for this problem
consists of 1370 overlapping tenday periods from January 2011 to mid June
2013.
What proportion of times did each fund decline 5 or more
times in a 10day period? Are these
proportions significantly different for a twotailed test at a 0.05 level of
significance?
Number of Declines in a
TenDay Period


Large Cap Fund

Large Cap
Fund

SmallCap
Fund

0

1

2

1

20

25

2

103

87

3

236

216

4

343

313

5

321

333

6

226

256

7

96

104

8

22

31

9

2

3

10

0

0

Total

1370

1370

Analysis:
First, calculate the
proportion of decline days 5 and over and the complement decline days fewer
than 5.
Daily Declines in Ten Day
Periods

% Declines in Ten Day
Periods


Large Cap Fund

SmallCap Fund


<5 Declines

703

643

0.51314

0.46934

>=5 Declines

667

727

0.48686

0.53066

1370

1370

1

1

Second, calculate the
tstatistic.
I like to calculate the standard error for the tstatistic
without pooling data to get a common proportion for >=5 declines over the
two funds.)
t= ((0.486860.53066)/ ((0.48686*0.51314/1370) +
(0.53066*0.46934)/1370))^{0.5}
This is equal to t= 2.30.
Third, compare the
area to the left of this tstatistic to 0.025, the size of the critical region defined
by our test stipulations.
Here is a useful normal distribution table on the web.
The area to the left of 2.3 is 0.0107.
This is small than 0.025.
We reject the null hypothesis that the proportion of 5 or
more decline days out of 10 is identical for the two funds.
The likelihood of a large number of decline days seems large
for the smallfund ETF.
A Finance Footnote: I have numbers for a third fund the total
stock market funds, which includes largecap, midcap and smallcap
stocks. This fund realized 706
occurrences with fewer than 5 of 10 down days and 664 with more than 5 declines
in 10 days. The number of large decline
periods is slightly smaller for the total stock market fund than for the
largecap fund. But this difference is
not significant.
Authors Note: I am putting lists of problems on probability
and statistics on the blog statistical resources.
The probability problems in this post will involve the
binomial distribtution.
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