Question

Refer to Exercise 13.74. Suppose that you have obtained a $95\%$confidence interval for each of the two differences, ${\mu }_1-{\mu }_2$and ${\mu }_1-{\mu }_3$. Can you be$95\%$ confident of both results simultaneously, that is, that both differences are contained in their corresponding confidence intervals? Explain your answer.

Short answer

The probability of both findings being contained in their respective confidence intervals at the same time is not exactly $0.95$.

Step-by-step solution

Given Information

Confidence Interval obtained for the each of the two differences ${\mu }_1-{\mu }_2$and ${\mu }_1-{\mu }_3=95\%$.

The probability that a population parameter will fall between a set of values for a particular proportion of the time is referred to as a confidence interval.

Explanation

No, it is not possible to be $95\%$sure in both outcomes at the same time, meaning that both discrepancies are contained inside their respective confidence ranges.

Let role="math" localid="1652193214871" $E_1$be the interval event with probability based on the difference ${\mu }_1-{\mu }_2$.

$P\left(E_1\right)=0.95$and the role="math" localid="1652193228446" $E_2$be the event of the interval based on the difference ${\mu }_1-{\mu }_3$with probability

$P\left(E_2\right)=0.05$

The simultaneous occurrence of role="math" localid="1652193390006" $E_1$and role="math" localid="1652193298728" $E_2$events is symbolized by $P\left(E_1\cap E_2\right)$, however the events $E_1$and role="math" localid="1652193407288" $E_2$are not independent in reality.

Using the multiplication formula, $P\left(E_1\cap E_2\right)=P\left(E_1\right)P\left(E_2\mid E_1\right)$can be calculated.

The information concerning $P\left(E_2\mid E_1\right)$is not mentioned in the circumstance. As a result, the events $E_1$and $E_2$cannot be calculated at the same time. Furthermore, $P\left(E_1\right)$has a value of $0.95$, and $P\left(E_2\mid E_1\right)$must be less than$0.95$.