What is the meaning of statistical significance and how valuable is this concept? A \(t\)-statistic is \(t=(b-c) / \operatorname{se}(b)\), where \(b\) is an estimate of a parameter \(\beta, c\) is the hypothesized value, and \(\operatorname{se}(b)\) is the standard error. If the sample size \(N\) is large, then the statistic is approximately a standard normal distribution if the null hypothesis \(\beta=c\) is true.

a. With a \(5 \%\) level of significance, we assert that an event happening with less than a one in 20 chance is "statistically significant," while an event happening with more than a one in 20 chance is not statistically significant. True or False?

b. Would you say something happening one time in 10 by chance \((10 \%)\) is very improbable or not very improbable? Would you say something happening one time in 100 by chance (1\%) is very improbable or not?

c. If we adopt a rule that in large samples, a \(t\)-value greater than 2.0 (in absolute value) indicates statistical significance, and we use Statistical Table 1 of standard normal cumulative probabilities, what is the implied significance level? If we adopt a rule that in large samples, a \(t\)-value greater than 3.0 (in absolute value) indicates statistical significance, what is the implied significance level?

d. Suppose that we clinically test two diet pills, one called "Reliable" and another called "More." Using the Reliable pill, the estimated weight loss is \(5 \mathrm{lbs}\) with a standard error of \(0.5 \mathrm{lbs}\). With the More pill, the estimated weight loss is \(20 \mathrm{lbs}\) with standard error \(10 \mathrm{lbs}\). When testing whether the true weight loss is zero (the null, or none, hypothesis), what are the \(t\)-statistic values? What is the ratio of the \(t\)-values?

e. If the drugs Reliable and More were equivalent in safety, cost and every other comparison, and if your goal was weight loss, which drug would you take? Why?