1. Tests of Significance
Section 10.2

2. Tests of Significance
Used to assess the evidence in favor of some claim.
Example: Test sweetener in cola to see if it loses its sweetness over time. Positive # means loss of sweetness. Results of ten tasters: 2.0, 0.4, 0.7, 2.0, -0.4, 2.2, -1.3, 1.2, 1.1, 2.3
X = Is this really loss of sweetness or just chance variation?
Suppose we know that σ = 1.

3. Sweetness Loss

4. Sweetness Loss
At x = 1.02 the p-value is

5. Steps in Test of Significance (HATS)
H- Hypotheses: State assumption, null hypothesis, H0, State what we suspect, the alternative hypothesis, Ha.
A- Assumptions: Check that a test can be conducted and what type of test should be used.
T- Test: Calculate Test Statistic (sample statistic used to estimate a population parameter) and see how likely it could happen by chance (P-value is probability).
S- Summarize: results with small p-values (< .05) rarely occur if null hypothesis were true – statistically significant.

6. To Compute P-value Ex: To find P(x ≥ sample mean),
Find standard deviation of sampling distribution.
Draw a picture.
Standardize x to find P(Z ≥ z), P-value.

7. To compute the z Test Statistic
z = (x - µ0) / (σ/√n)
If H0: µ = µ0
Ha: µ > µ0 find P(Z ≥ z)
Ha: µ < µ0 find P(Z ≤ z)
Ha: µ ≠ µ0 find P(Z ≤ z) or P(Z ≥ z)
Draw each situation.

8. Example:
Mean systolic blood pressure for males is reported to be 128 with a st. dev. of 15. A sample of 72 executives have a mean of Is this evidence that the executives have a different mean blood pressure from the general public?
H H0: µ = 128; Ha: µ ≠ 128 Draw!!!
A Assumptions?
T Test Statistic Z and P-value
S Summarize Results

9. Executives Example Test Statistic and P-value
Summary: 27% of the time a sample would differ from the population in this way. This is not good evidence that the executives differ from others.

10. Significance Level
Sometimes we have a predetermined value of p, written as α, alpha.
If the p-value is as small as, or smaller than, α, then we say “The data are statistically significant at level α.”
If Z ≥ z* then reject null hypothesis.
Never accept null hypothesis. Reject or fail to reject.

11. Example: Test with given α
Concentration of active ingredient is reported to be .86 with a st. dev. of A sample of 3 measurements have a mean of Is there significant evidence at the 1% level that µ ≠ .86?
H H0: µ = .86; H0: µ ≠ .86 Draw!!!
A Assumptions?
T Test Statistic Z (Use table C to compare to 99% z*)
S Summarize Results

12. Example Test Statistic z
Summary: z was much larger than z* so we reject the hypothesis that the concentration is .86.

13. Significance Tests Steps (HATS):
H: ID Population and parameter of interest. State null and alternative hypothesis.
A: Choose appropriate inference procedure. Verify assumptions for using procedure.
T: Carry out Inference Test:
Make a sketch!
Calculate test statistic.
Find P-value.
S: Interpret results and summarize in the context of the problem.