Significance Tests (notes)

Notes

• The significance test we look at are 5% or 10%
• This is the probability (0.05 or 0.10) that a result is "too unusual"
• A null hypothesis is when H0 is unchanged
• An alternative hypothesis is when a change is suggested (H1)
• If H0 : p=p1 , the possible situations for H1 are: p>p1 , p<p1 , p≠p1
• p>p1 and <p1 are both 1 tailed , while p≠p1 is 2 tailed
• The test statistic is p( ) of the result from data collected

Example 1

• Declan believes over 20% of students eat a least 5 portions of fruit and veg everyday. Declan asks the 25 students in his other statistics classes and 8 of them say they eat at least 5 portions of fruit and veg every day.
• Assuming that these 25 students may be considered to be a random sample, carry out a hypothesis test at he 5% significance level to investigate whether Declan's belief is supported by this evidence
• H0 : p = 0.2 (20%) , H1 : p > 0.2 (alternate hypothesis, which is later not needed) , X = number of students eating 5 fruit/veg , X~B(25,0.2)
• p(x≥8) = 1-p(x≤7) = 0.109 , 0.109 > 0.05 (5%) therefore insufficient evidence at the 5% to suggest H0 is false
• We accept H0 therefore the result suggests the percentage of students eat at least 5 fruit/veg per day is 20%

Example 2

• The probability of recovering from swine flu after taking a drug called frazzles is thought to be 0.85. From a trial of 30 people taking the drug, 29 recover.
• Test at the 10% significance level whether the probability is different to 0.85.
• H0: p=0.85 (85%) , H1: p≠0.85 (as "different to 0.85") , X number of frazzle recovery , X~B(30,0.85)
• p(x≥29) = 1-p(x≤28) = 0.048. Now as we have p≠0.85 , this is 2 tailed so we must divide 10% by 2 to get 5%. If we calculate the mean using np we get 30 x 0.85 = 25.5 , and 29 is to the right of this so we can say 0.048<0.05 , therefore there is sufficient evidence at the 10% level to suggest H0 is false reject , so we will reject H0 in favour of H1.
• This result suggests the chance of recovering from swine flu using frazzles is not 0.85.