For each problem, select the best response.
(a) does 30 minutes of aerobic exercise each day provide significant improvement in mental performance? to investigate this issue, a researcher conducted a study with 150 adult subjects who performed aerobic exercise each day for a period of six months. at the end of the study, 200 variables related to the mental performance of the subjects were measured on each subject and the means compared to known means for these variables in the population of all adults. nine of these variables were significantly better (in the sense of statistical significance) at the α=0.05α=0.05 level for the group that performed 30 minutes of aerobic exercise each day as compared to the population as a whole, and one variable was significantly better at the α=0.01α=0.01 level for the group that performed 30 minutes of aerobic exercise each day as compared to the population as a whole. it would be correct to conclude
a. that these results would have provided very good statistical evidence that 30 minutes of aerobic exercise each day provides some improvement in mental performance if the number of subjects had been larger. it is premature to draw statistical conclusions from studies in which the number of subjects is less than the number of variables measured.
b. that there is very good statistical evidence that 30 minutes of aerobic exercise each day provides some improvement in mental performance.
c. that there is very good statistical evidence that 30 minutes of aerobic exercise each day provides improvement for the variable that was significant at the α=0.01α=0.01 level. we should be somewhat cautious about making claims for the variables that were significant at the α=0.05α=0.05 level.
d. none of the above.
(b) the nicotine content in cigarettes of a certain brand is normally distributed with mean (in milligrams) μμ and standard deviation σ=0.1σ=0.1. the brand advertises that the mean nicotine content of their cigarettes is 1.5, but you believe that the mean nicotine content is actually higher than advertised. to explore this, you test the hypotheses h0: μ=1.5h0: μ=1.5, ha: μ> 1.5ha: μ> 1.5 and you obtain a p-value of 0.052. which of the following is true?
a. there is some evidence against h0h0, and a study using a larger sample size may be worthwhile.
b. you have failed to obtain any evidence for ha. ha.
c. this should be viewed as a pilot study and the data suggests that further investigation of the hypotheses will not be fruitful at the α=0.05α=0.05 significance level.
d. at the α=0.05α=0.05 significance level, you have proven that h0h0 is true.
(c) an engineer designs an improved light bulb. the previous design had an average lifetime of 1200 hours. the new bulb had a lifetime of 1200.2 hours, using a sample of 40,000 bulbs. although the difference is quite small, the effect was statistically significant. the explanation is
a. that the sample size is very large.
b. that the mean of 1200 is large.
c. that new designs typically have more variability than standard designs.
d. all of the above.