=5.5 days reflect a real change in absenteeism or could we easily get the outcome by chance alone?Math 37 - Lecture 18
Significance Tests (6.2, 8.1)
Confidence Intervals - Estimate population parameter.
Tests of Significance - Make a conclusion about the population.
Does data refute our hypothesis?
Reasoning:
Example In the 1980’s, many companies experimented with "flex-time", allowing employees to choose their schedules within broad limits set by management. Among other things, flex-time was supposed to reduce absenteeism. Suppose one firm knows that in the past few years, employees have averaged 6.3 days off from work. This year, the firm introduces flex-time. Management chooses a simple random sample of 100 employees to follow, and at the end of the year these employees average 5.5 days off from work and the standard deviation is 2.9 days. Does flex-time reduce absenteeism?
- What do we know?
- What are we trying to decide?
- What is the parameter of interest?
- Does the sample result =5.5 days reflect a real change in absenteeism or could we easily get the outcome by chance alone?
If there has not been a change, what should the sampling distribution of look like?
Is this result unusual if there hasn't been a change due to flex-time?
How measure this?
Define parameter Let m=average number of days absent with flex-time
1. State the possibilities
Null Hypothesis no change, e.g. m=6.3 (mean stayed the same)
Alternative Hypothesis: is a difference, e.g. m<6.3 (improvement)
Observe =5.5
How far is this observation from the value claimed in the null hypothesis? How many standard deviations are we off from m=6.3? If the null hypothesis really is true, how likely are we to have observed such a data value? If very unlikely, we have evidence against the null hypothesis.
2. Assume the null hypothesis is true: m=6.3
Is the value of surprising if the null hypothesis is true?
What indicates the values of we expect to see when m=6.3?
3. Calculate a test statistic, z0
How do we measure if the observation is unusual? How often it occurs? Standardize the observed statistic.
What are the chances of getting a test statistic as extreme or more extreme than the one we observed if the null hypothesis is true?
4. Calculate the p-value=the probability of observing this value
P(Z<zo if null hypothesis is true)=p-value=observed significance level. If the probability of such a value is small (i.e. if the observed significance level/p-value is small) we have strong evidence against null hypothesis.
5. Make a conclusion
If the p-value is too small, we reject the null hypothesis and go with what the alternative hypothesis said. If the p-value isn't too small we fail to reject the null hypothesis and go with what the null hypothesis said. In this example, the p-value was quite small so we reject the null hypothesis that the absenteeism didn't chance and conclude that the mean did become smaller after switching to flex.
Example Punxsutawney Phil has correctly predicted whether or not winter would last 6 more weeks 10 times out of 17 winters since 1980. Can this be a result of pure chance? Phil has predicted more winter 100 times in 112 years of forecasting - can this be a result of pure chance?