Trials and probability

Determining whether rain on

consecutive days is an independent

phenomenon

Remember what independence means: Trials are independent if each trial
is unaffected by all previous trials.

Question: How can we tell whether precipitation on a particular day is
affected by precipitation on previous days?

Answer: Let’s assume there is no effect. Then, If it rains on a given
day, say Wednesday, the chance that it will rain on the following day
(Thursday) should be no different that if it did not rain on Wednesday.

SO what we need to do is create a data log of days and catalogue whether
or not it rained on each day. Then we must figure out two things:

What is the probability that it rained on a particular day, given that
it rained on the previous day?

What is the probability that it rained on a particular day , given that
it did not rain on the previous day?

If there two probabilities are roughly the same, then we can conclude
that rain on consecutive days is an independent phenomenon. If the
probabilities are significantly different, then we can conclude that
rain on the consecutive days in a dependent phenomenon.

Example:

Day 1 2 3 4 5 6 7 8 9 10

Rain? Y/N N N Y Y Y N N N N Y



Look at the days when it rained: 3, 4, 5, 10

What are the chances that it rained on the next day? 2 out of 3, or 67%

Look at eh days when it did not rain: 1, 2, 6 7, 8, 9,

What are the chances that it rained on the next day? 2 out of 6, or 33%

The sample size in this case is not sufficiently large to make a
definite determination, but you can see that in this case it seems that
it is more likely to rain if it rained the previous day, and hence we
would conclude that this is not an independent phenomenon.

Your task is to do a similar experiment using actual weather data. Use a
sample sixe of atleast 180 days. Remember, the larger the sample size,
the more accuratge results you’ll get. You can find this data from the
following website: HYPERLINK
"http://www.almanac.com/watherhistory/index.php"
www.almanac.com/watherhistory/index.php

Steps:

Collect the data, citing your source and time span you used

Find p-hat(R), the proportion of days on which it rained given that it
rained the pervious day.

Find p-hat (NR) the proportion of days on which it rained given that it
did not rain the previous day.

Construct confidence intervals for both p-hat(R) and P-Hat(NR) (you can
chose level of confidence)

Draw a conclusion as to whether rain on consecutive days is an
independent phenomenon. State your conclusion by saying there is(or is
not) enough of a discrepancies to support the claim that rain on
consecutive days is a dependent phenomenon.