Question 15.3: Consider the i.i.d. r.v.’s X1, . . . , X20 with continuous d......

Using formula (11),
P(Y_{i}\leq x_{p}\leq Y_{j})=\sum\limits_{k=i}^{n}{\binom{n}{k}}p^{k}(1-p)^{n-k}-P(Y_{j}\leq x_{p})

~~=\sum\limits_{k=i}^{n}{\binom{n}{k}}p^{k}(1-p)^{n-k}-\sum\limits_{k=j}^{n}{\binom{n}{k}}p^{k}(1-p)^{n-k}

~~=\sum\limits_{k=i}^{j-1}{\binom{n}{k}}p^{k}(1-p)^{n-k}.           (11)

we obtain the coverage probabilities listed in Table 15.1 for several confidence intervals for the median x_{0.50} and the first quartile x_{0.25}. For the calculation of coverage probabilities for confidence intervals for the third quartile x_{0.75}, we employ the following formula, which allows us to use the Binomial tables; namely,
\sum\limits_{k=i}^{j-1}{\binom{20}{k}}(0.75)^{k}(0.25)^{20-k}=\sum\limits_{r=20-j+1}^{20-i}{\binom{20}{r}}(0.25)^{r}(0.75)^{20-r}.