Example A dog-food manufacturer sells "50-lb" bags of dog food. Suppose the weights of the bags are Normally distributed with standard deviation .84lb. You randomly select 75 bags and find that =50.11 lb. Do you think there is significant evidence that the actual mean weight of all "50-lb" bags of this dog food differs from the advertised weight of 50 lb.?

Alternative Hypothesis (H_a) - What counts as evidence against H₀

One Sided - Only care about deviations from H₀ in one direction

Example - Did flextime reduce absenteeism? Was % wins too large?

Two Sided - Look for evidence in either direction

Example - Did absenteeism change? Does the % differ?

Two sided hypotheses tests consider values of statistic that are

either too small or too large as evidence against the null hypothesis.

Use if don’t know/care ahead of time which way change may go.

Calculating the p-value: How you state the alternative determines how you calculate the p-value (what counts as evidence).

Example If using as test statistic:

One-sided p-value vs. two-sided p-value

Interpretation Indicates the probability that the test statistic takes on a value at least as extreme as the one observed if H₀ is true.

2. Specifying a Level of Significance, a (optional)

Can set a ahead of time to specify how strong you want the evidence to be. Tells you how unusual the observation needs to be for you to consider it strong enough evidence to reject H₀.

If p-value< a, we reject H₀ (say result is statistically significant)

If p-value>a, we fail to reject H₀

Preference for reporting p-values: Tells you how strong the evidence is, often more important than if reject.

For Fixed Significance Levels: We can decide if the result is "statistically significant" without calculating the p-value by finding the critical value z* that corresponds to the significance level. If z₀ is more extreme than z*, we say z₀ is in the "rejection region".

One sided:

Two sided:

Look familiar? A level a two-sided significance test rejects a hypothesis Ho:m=m₀ exactly when the value m₀ falls outside a confidence interval with confidence level 1-a.

Example Construct a 95% confidence interval for m, the true weight of the bags based on this sample result. From this interval - for which values of m₀ would we fail to reject H₀?

Example In a series of cases argued in the south between 1960 and 1980, expert witnesses presented the case for racial bias in jury selection. Panels of jurors are theoretically drawn at random from a list of eligible citizens. However, in southern states in the 50's and 60's few African Americans were found on jury panels, so some defendants challenged the verdicts. Evidence: 50% of eligible citizens were African American. On an 80-person panel of potential jurors, only four were African American. Can this be the result of pure chance?

Example Diet colas use artificial sweetners to avoid sugar. Colas with artificial sweeteners gradually lose their sweetness over time. Sweetness losses: 2.0 .4 .7 2.0 -.4 2.2 -1.3 1.2 1.2 2.3

Is this different due to chance or does it reflect a real loss in sweetness?

Example In the last 13 World Series, the team with home field advantage won 12 times. Similarly for the last 43 of 73, are either of these facts convincing evidence of a higher likelihood of winning?