STAT 1301 Tests of Significance about Averages

Concern has been mounting that SAT scores are falling. 3 years ago -- National AVG = 955 Random Sample of 200 graduating high school students this year : AVG = 935 SD = 100 Question: Have SAT scores dropped ? Procedure: Determine how “extreme” or “rare” our sample AVG of 935 is if population AVG really is 955.

We must decide: The sample came from population with population AVG = 955 and just by chance the sample AVG is “small.” OR We are not willing to believe that the pop. AVG this year is really 955. (Conclude SAT scores have fallen.)

Common Terminology if P- value < 5% -- “ statistically significant” -- “highly statistically significant” if P- value > 5%. -- Typically, “do not reject” hypothesized model

Significance Test Write-up Procedure 1. State the null hypothesis 2. State the alternative hypothesis 3. Give the test statistic used and the calculations 4. State the P-value 5. State the conclusion in the language of the problem

Hypothesis Testing Logic Null -- “nothing new is happening” Alternative -- what we “want” to show Collect data If - data supports alternative - outcomes this extreme in support of the alternative could occur very rarely (< 5% of the time) when null is true Then we reject the null.

In our case (school district data): - P-value = 11.5% - not rare enough - do not reject null - doesn’t mean we believe AVG=100 (we simply do not have enough evidence to reject it.)

SO: If we gather really strong evidence against the null, we reject it and we feel good that our decision to reject the null is correct If we don’t gather sufficiently strong evidence against the null, we do not reject it and we realize that the null still might not be true so we say “do not reject the null” instead of “accept the null”