Significance Testing

1
Significance Testing

• Statistical testing of the mean (z test)

2
Binomial Distribution

• Mathematicians have figured formulas to estimate
  long run relative frequencies for simple events,
  like how many heads will appear for a given
  number of coin tosses. The binomial is one such.

Recall dice
Number of heads in 10 flips of a coin.
3
Normal Distribution

• We have already figured percentages of the
  normal. Percentages of the normal correspond to
  probabilities of finding individual cases in the
  distribution. The sampling distribution of the
  mean is normal if N is large.

Middle 95 percent from going up and down 1.96 SDs
from the mean.
4
Significance Testing 1

• Significance testing is a what if game.
• We make an assumption, and ask what will happen
  if the assumption is true. Assumption is null
  hypothesis.
• Significance testing is based on probabilities
  that come from the what if scenario (from the
  null hypothesis).

• What if the true mean height of students at USF
  is 66 inches and SD is 5 inches? What if we draw
  people from USF 100 at a time and plot the means?
• We can figure the sampling distribution from
  these assumptions and figure probabilities.
• Rejection region is place that is unlikely to
  occur if null is true.

5
Significance Testing 2

• Given
• Result 1 Sampling distribution of mean is
  normal (mu 66).
• Result 2 Standard error of the mean is

Rejection Region
Rejection Region
What is a rejection region?
6
Review

• Suppose population mean is 500, population SD is
  100 (SAT data), and sample size is 100.
• Draw sampling distribution of means.
• What is the shape of this distribution?
• What is the mean of this distribution?
• What is the standard deviation of this
  distribution?
• Find, mark, and label the rejection regions.

7
Review
Upper 5001.96(10)519.6
Lower 500-1.96(10) 480.4
RR gt 519.6 RR lt 480.4
RR
RR
Shape is normal, mean is 500, SD is 10.
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Significance Testing 3

• Establish what if
• Collect sample data.
• Examine probability of sample result given the
  null.
• If probability is low, say that null is false,
  i.e., reject the null hypothesis.
• This is a significance test. If we reject the
  null, we say result is statistically significant.
• Significance testing lets us make decisions about
  populations from sample data.

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Example

• Mean beers at Skippers Smokehouse?
• Null (what if)
• Data from Skippers
• Derive

Reject the null. Result is significant.
Observed data are very unlikely if null is true.
Null must be false. Lots of beer at Skippers.
Note data are fictitious.
10
Review

• We want to know if a workbook helps with learning
  stats. We know from past classes that students
  average 75 percent on the final with a SD of 5.
  Our new class of 225 has a mean of 78. Did the
  workbook help? (Hint sqrt(225) 15.)

11
Review
Upper 751.96(.33) 75.65. This is far below
78. The workbook helps.
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Definition

• The term probability refers to the long run
• 1 Frequency of outcome
• 2 Odds ratio
• 3 Relative frequency
• 4 Rolling of dice

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Definition

• The calculation of probabilities in hypothesis
  testing rests upon assumptions described in the
  ______.
• 1 Alternative hypothesis
• 2 Null hypothesis
• 3 Sampling distribution
• 4 Standard error

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Definition

• The rejection region is the place in the sampling
  distribution that is _____
• Close to the mean
• Obtained when the result is not significant
• Very unlikely if the null hypothesis is true
• Visited by losers