Final Project

• Some details on your project

– Goal is to collect some numerical data pertinent to some question and analyze it using one of the statistical tests we’ve discussed in class

– You will be graded on all aspects of the task from the nature of the question to the execution of the statistical test

Final Project

• Some examples:– Does the price of oil correlate with the price of gasoline?

• Approach: record daily price of oil and the price of gas at some gas station over several weeks and run a correlation

– Is Calgary colder/windier/rainer than Edmonton• Collect data from Environment Canada’s web site

– Do Canadians score more than other NHL players?• Collect data from any sports section or website

Final Project

• Guidelines:– Use readily available observational data

• Don’t run an experiment unless you check with me first!!!

– Keep questions simple and straightforward• Get your idea checked by Farshad before you proceed

– Plan to do your project with Excel or some stats program• Turn in the data, the relevant statistics, and one or two

sentences explaining your question and the answer - should fit on one page.

Some Review

• A population is a really big bunch of numbers

Some Review

• A population is a really big bunch of numbers

• A sample is some of the numbers from a population

Some Review

• All sets of numbers have a distribution– The population has a mean– A sample has a mean that is probably

similar but not necessarily the same as the population

Some Review

• All sets of numbers have a distribution– The population has a standard deviation– A sample has a standard deviation that is

probably similar but not necessarily the same as the population

Some Review

• If we think in terms of standard deviation, we can know things like whether or not a single number is very different from the mean of a population

Some Review

• But often we’re not interested in single numbers - we’ve collected a sample and computed a mean

• That mean comes from a population of sample means (you just happened to pick one of them)

• The mean of the distribution of sample means is the mean of the population

• The standard deviation of the sample means is the standard error

Some Review

• If we think in terms of standard errors, we can know things like whether a particular mean is very different from the mean of a population

Keep these ideas straight

• If we think in terms of standard deviation, we can know things like whether or not a single number is very different from the mean of a distribution

• If we think in terms of standard errors, we can know things like whether a particular mean is very different from the mean of a population

€

Zx =x −μ x

σ x

€

zi =x i − x

Sx

Some Review

• We use the Z table to look up the probability that a particular Z score came from any normal population

Some Review

• We use the Z table to look up the probability that a particular Z score came from any normal population

• Since the population of sample means is normal (Central Limit Theorem), we can use the same Z table to look up the probability that a sample mean came from a population with a particular mean

Now a Real Example

• Break into groups of 10

• Write down your heights in inches

• Compute the mean of your n=10 sample

• Compute the standard deviation

• Hand it all in to Fraser

Critical Z Value

• In our examples we’ve been testing the hypothesis that one sample has a mean that is higher (or lower) than a population mean

Critical Z Value

• In our examples we’ve been testing the hypothesis that one sample has a mean that is higher (or lower) than a population mean

• Let’s turn this around a bit…let’s work backwards

Critical Z Value

• How much bigger would a sample mean have to be so that there’s only a 5% chance that it came from a particular population?

Critical Z Value

• How much bigger would a sample mean have to be so that there’s only a 5% chance that it came from a particular population?

Gaussian (Normal) Distribution

0

0.1

0.2

0.3

0.4

0.5

0.6

-4 -3 -2 -1 0 1 2 3 4

score

probability

95%

This is the alpha = .05 threshold

What Z score?

5%

Critical Z Value

• This is sometimes called the critical Z value or

€

Zcrit (one − tailed) =1.64

Directional vs. Bidirectional Tests

• In our examples we’ve been testing the hypothesis that one sample has a mean that is higher (or lower) than a population mean

• We call this a directional or “one-tailed” test

• What does that one-tailed bit mean !?

Directional vs. Bidirectional Tests

• We were checking to see if our sample had a mean far enough into the positive tail of the distribution and ignoring the negative tail

Directional vs. Bidirectional Tests

• Often we haven’t made a directional hypothesis, but have simply predicted “a difference”

Directional vs. Bidirectional Tests

• Often we haven’t made a directional hypothesis, but have simply predicted “a difference”

• In that situation, we are twice as likely to make a Type I error: the sample mean could, by chance, be in either tail !

Directional vs. Bidirectional Tests

• What would the critical Z value be so that there is a 5% chance that a mean is beyond it in either direction?

Directional vs. Bidirectional Tests

• What would the critical Z value be so that there is a 5% chance that a mean is beyond it in either direction?

Gaussian (Normal) Distribution

0

0.1

0.2

0.3

0.4

0.5

0.6

-4 -3 -2 -1 0 1 2 3 4

score

probability

95%

This is the alpha = .05 threshold

What Z score?

2.5%2.5%

Directional vs. Bidirectional Tests

• Thus:

€

Zcrit (two− tailed) = + −1.96