10.2 Tests of Significance
Significance level (  ): used to compare the p-value with a fixed value which we regard as a decisive factor as to whether a value will be statistically significant.
p-value: the probability that x will take on a value as extreme or more extreme than the actual observed value (
(If the P-value is as small or smaller than
 )

 then we say that the data are statistically significant at level
)

Diet colas which use artificial sweeteners were studied to observe whether they lose sweetness over time. There were ten taste testers which rated the amount of sweetness lost on a scale of zero to ten. Here were the results:
2.0
0.4
0.7
2.0
-0.4
2.2
-1.3
1.2
1.1
2.3
Is this good evidence that the colas lost sweetness?
Average loss = 1.02
Is this just by chance or is there a true loss of sweetness?

The TEST:
State the null hypothesis is no change
H
0
:   0
H
0 which states that there
(no loss in sweetness)
State the alternative hypothesis that there is change
H a
:   0
H a which states
(loss in sweetness)
If we assume the null hypothesis is true, would x  1 .
02 be a large enough mean from the sample to conclude that the null hypothesis must be false?
Assume that would be a value: x 
1
.
3

10
 1 then the standard deviation
= .316 which means if   could occur “just by chance”
(approx. one s.d. from mean)
0 then x  1 .
02 is unlikely to occur “just by chance”
(approx. three s.d. from mean)

1) How likely are these values (.3 and 1.02)?
2) How much convincing do you need?
1) Calculate P-values
.
3 
.
316
0
 .
95 .
1711
17% of all samples will give a result of .3
or greater “just by chance”
1 .
02 
.
316
0
 3 .
23 .0006
.06% of all samples will give a result of
1.02 or greater “just by chance”
2) Rule of Thumb: use a level of .05 to determine if the P values are statistically significant (happening “just by chance” is rare)
.1711 is not statistically significant
.0006 is statistically significant

Mr. X claims that the average score for his class should be a bit higher than 90. Test Mr.
X’s claim using the following data.
The scores on an AP Stat test were:
92 93 95 79 89 91 92 85 93 92
State the hypothesis: (use correct notation!)
Find z-score: (show correct mechanics)
Find p-value: (calculator)
State a conclusion:
(BE SPECIFIC TO THE PROBLEM!!!!!!!!!!)