If we have a large sample of data, then using critical values from the standard normal distribution for constructing a \(\boldsymbol{p}\)-value is justified. But how large is "large"?

a. For a \(t\)-distribution with 30 degrees of freedom, the right-tail \(p\)-value for a \(t\)-statistic of 1.66 is 0.05366666 . What is the approximate \(p\)-value using the cumulative distribution function of the standard normal distribution, \(\Phi(z)\), in Statistical Table 1? Using a right-tail test with \(\alpha=0.05\), would you make the correct decision about the null hypothesis using the approximate \(p\)-value? Would the exact \(p\)-value be larger or smaller for a \(t\)-distribution with 90 degrees of freedom?

b. For a \(t\)-distribution with 200 degrees of freedom, the right-tail \(p\)-value for a \(t\)-statistic of 1.97 is 0.0251093 . What is the approximate \(p\)-value using the standard normal distribution? Using a two-tail test with \(\alpha=0.05\), would you make the correct decision about the null hypothesis using the approximate \(p\)-value? Would the exact \(p\)-value be larger or smaller for a \(t\)-distribution with 90 degrees of freedom?

c. For a \(t\)-distribution with 1000 degrees of freedom, the right-tail \(p\)-value for a \(t\)-statistic of 2.58 is 0.00501087 . What is the approximate \(p\)-value using the standard normal distribution? Using a two-tail test with \(\alpha=0.05\), would you make the correct decision about the null hypothesis using the approximate \(p\)-value? Would the exact \(p\)-value be larger or smaller for a \(t\)-distribution with 2000 degrees of freedom?