1. A researcher hypothesizes that cochlear implant use results in better speech
intelligibility for deaf talkers than hearing aids. The hypothesis is tested by comparing
the intelligibility of words spoken by 15 cochlear implant users with 17 otherwise
comparable deaf speakers who use a hearing aid. Results were as follows: xˉ Implant =
79.6; xˉ HearingAid = 72.6; sxˉ 1-xˉ 2 = 6.2. Are these two means significantly different?
Method:
(1) t = xˉ 1-xˉ 2/ sxˉ 1-xˉ 2 = 79.6 – 72.6 / 6.2 = 1.12903
(2) df = n1+n2-2 = 15+17-2 = 30
(3) The hypothesis is directional (see sentence 1), also known as one-tailed
(4) A table look-up at df = 30; 0.05 one-tailed (directional) shows a critical value of
1.697. The value we computed must be at least as large as this threshold value in
order to reject the null. The t value we computed of 1.12903 is not large enough.
(5) Decision: Fail to Reject the Null, meaning: (a) the difference is not significant
and, equivalently, (b) there is too high a probability that the mean difference we
obtained was due to sampling error. Note that we have not proven the Null.
2. A researcher hypotheses that children who receive ear training will show
improvements in articulation that are greater than those who do not receive ear
training. The hypothesis is tested by comparing articulation improvement in 12
children who received ear training versus 12 otherwise comparable children who did
not receive ear training. Results were as follows: xˉ EarTraining = 10.6; xˉ NoEarTraining = 4.9;
sxˉ 1-xˉ 2 = 2.2. Are these two means significantly different?
Method:
(1) t = xˉ 1-xˉ 2/ sxˉ 1-xˉ 2 = 10.6 – 4.9 / 2.2 = 2.59091
(2) df = n1+n2-2 = 12+12-2 = 22
(3) The hypothesis is directional (see sentence 1), also known as one-tailed
(4) A table look-up at df = 22; 0.05 one-tailed (directional) shows a critical value of
1.717. The value we computed must be at least as large as this threshold value in
order to reject the null. The t value we computed of 2.59091 is larger than this
threshold value.
(5) Decision: Reject the Null, meaning: (a) the difference is statistically significant
and, equivalently, (b) there is low probability (less than 5%) that the mean
difference we obtained was due to sampling error.
(6) What is the chance that our decision to reject the Null is wrong? Less than 5%.