Question

Referring to Example \(7.9 .5\) of this section, determine \(c\) so that $$P\left(-c

Short answer

To get the 95% confidence interval for \(\theta\) given \(T_{2}=t_{2}\) we would calculate \(c\) then construct the interval as everything between \(T_{1}-c\) and \(T_{1}+c\).

Step-by-step solution

Applying the definitions of conditional probability

The given relation can be interpreted based on the definitions of conditional probability \(P\left(-c<T_{1}-\theta<c \mid T_{2}=t_{2}\right)=0.95\). This states the boundary conditions that the difference of \(T_{1}\) and \(\theta\) lies within \(c\) given \(T_{2}=t_{2}\). And this boundary is given to us as 95 % probability.

Transforming the inequalities into a more manageable form

Rewrite the probability as \(P\left(T_{1}-c <\theta< T_{1}+c \mid T_{2}=t_{2}\right)=0.95\). This is the standard form of a two-sided confidence interval - the parameter \(\theta\) lies between \(T_{1}-c\) and \(T_{1}+c, given T_{2}=t_{2}\) with 95 % probability.

Constructing the confidence interval

This problem doesn't provide functional relationships between \(\theta, T_{1}, \) and \(T_{2}\), it's not possible to provide a specific numeric confidence interval or value of \(c\). In a real world scenario, you would use your estimate of \(T_{1}\) and your calculated value of \(c\), then your 95% confidence interval for \(\theta\) would be \((T_{1}-c, T_{1}+c)\) given \(T_{2}=t_{2}\) .

Key concepts

Conditional Probability and Its Importance

Conditional probability is a crucial concept in statistics that helps us assess the likelihood of an event occurring, considering that another related event has already occurred. It's expressed as \( P(A | B) \), meaning the probability of event \( A \) given that \( B \) has occurred. This concept helps in understanding many statistical phenomena and is frequently used in various fields
like finance, medicine, and even daily decision-making situations. In this exercise, we use conditional probability to analyze how closely two variables, \( T_1 \) and \( \theta \), relate to each other when the condition \( T_2 = t_2 \) is known.
Understanding the relationship helps us make more accurate predictions and construct confidence intervals.

Understanding Confidence Level

In statistical analysis, the confidence level indicates the probability that a calculated confidence interval contains the true population parameter. In simpler terms, if we perform an experiment 100 times and calculate 100 confidence intervals, a 95% confidence level suggests that 95 of those intervals will contain the true parameter.
This concept provides a form of assurance about the estimates we derive from data. In our exercise, we focus on a 95% confidence level. This means we are 95% sure that the true value of \( \theta \) lies within the interval constructed using the observed data
and the value of \( c \).

Two-Sided Confidence Interval Demystified

A two-sided confidence interval is a statistical range calculated from the data so that the parameter of interest, in this case \( \theta \), falls within this range with a specified probability already set, which is often at a certain confidence level. Unlike one-sided confidence intervals, which only give boundary in one direction, a two-sided interval provides boundaries on both sides, allowing us to say with a certain level of confidence
that the actual parameter lies between these limits.
In the problem, this is represented as \( P(T_1 - c < \theta < T_1 + c \mid T_2 = t_2) = 0.95 \), meaning the true parameter \( \theta \) falls within \( T_1 - c \) and \( T_1 + c \) with 95% confidence. Two-sided confidence intervals are particularly useful when you need to understand the range within which a parameter is likely to lie rather than a single-point estimate.
They are pivotal in many scientific studies that seek balanced estimates without bias towards one direction.