More Stats Today

• Confidence Intervals
• Margin of Error
• Hypothesis Testing
• One and Two Tail
• p-value

How many samples do I need?

• the plus or minus margin m = z*(sigma/n^1/2)
• n=(z*sigma/m)²

• Assume sigma is 3 for this example.
• want to be accurate to 2 pounds (plus or minus 2) with 95% confidence
• z=1.96
• n=(1.96*3/2)²=8.6=9
• Let's say he only wants to be within 3 pounds
• n=(1.96*3/3)²=3.8=4

Properties of Confidence Intervals

• What is the relationship between
• 50% confidence interval
• 95% confidence interval
• 99% confidence interval
• Which is widest?
• Are they always symmetric?
• Confidence interval and sample size
• You can use your desired width of a confidence interval to decide how many observations you need.

How accurate is the confidence interval?

• If data are well collected
• The true mean will fall within in the interval with probability p.
• For a 95% confidence interval, the true mean will fall in the interval 95% of the time
• Sometimes it will not

Potential Problems

• Bad Experiment or Survey
• Data might come from a poor sample
• Bias will influence the mean of the sample
• Outliers in the data can disrupt estimate of the mean
• The standard deviation of the population has to be known
• We will talk about what to do when the standard deviation is not known soon.

Hypothesis Testing

• Often we want to know whether our data reveal a reliable effect.
• We can test this possibility using the techniques we have described
• This process is called statistical inference or hypothesis testing
• Suppose Kelloggs claimed that there were an average of 250 raisins in each box of Raisin Bran.
• How could we test this claim?

Null Hypothesis

• Start by determining the null hypothesis
• A state of affairs against which we are comparing
• In our case
• H₀:  mu = 250
• H₁:  mu < > 250
• The alternative hypothesis (H₁) is always stated relative to the null hypothesis.

One Tail vs Two Tail

• Two Tail
• Preferable
• H₁ is simply different than H₀
• One Tail
• Avoid in general
• More power (but don't cheat)
• can't look at the data before choosing one tail
• can miss big unexpected effects

P-value

• We can use the normal distribution to determine the likelihood of an observed outcome.
• Calculate the z-score of the observed mean relative to the sampling distribution with the mean in the null hypothesis.
• Find the probability of a point as extreme or more extreme than that (in either direction)

Calculating the p-value

• In our example:
•  mu₀ = 250
•  z (263) = (250 - 263) / 2.6 = -5.00
• 2.6 is the standard error of the mean (from previous lecture)
•  probability associated with a z-score of 5.00 is less than .0001

Pick a significance level

• How sure do you want to be in your answer?
• The probability associated with your sureness is called the a level
• In science, the alpha level selected is usually .05
• If the probability of observing a result is as extreme or more extreme than the alpha level, then reject the null hypothesis

In our example

• The probability associated with a mean of 263 is less than .0001.
• .0001 is smaller than .05
• Reject the null hypothesis
• That is, reject the statement that the population mean equals 250

Relationships to Confidence Intervals

• Another way to do the same test
• Formulate a confidence interval around the observed mean
• Use it to set the width
• If the mean from H₀ is inside the interval
• Do not reject H₀
• If the mean from H₀ is outside the interval
• Reject H₀

What about smaller alpha levels?

• Why do we use an a level of .05?
• There is a 5% chance that we will reject H₀ when we should not have (Type I error).
• If we used an a level of .01 there would only be a 1% chance that would happen.
• As the a level gets larger, there is a greater chance that we will decide not to reject H₀ when we should have
• Type II error
• It could be a problem to make a test too conservative
• The a level of .05 is a good compromise
• The Power of a test is 1 minus the probability of a Type II error.