Problem 12 - An Introduction to Mathematical Statistics and Its Applications

Question

Experience has shown that only \(\frac{1}{3}\) of all patients having a certain disease will recover if given the standard treatment. A new drug is to be tested on a group of twelve volunteers. If the FDA requires that at least seven of these patients recover before it will license the new drug, what is the probability that the treatment will be discredited even if it has the potential to increase an individual's recovery rate to \(\frac{1}{2} ?\)

Step-by-Step Solution

Answer

The probability that the treatment will be discredited even if it has the potential to increase an individual's recovery rate to 1/2 is the sum of the binomial probabilities for 6 or fewer successful trials out of 12, with a success probability of 0.5 for each trial.

Step 1: Understanding The Problem

The problem requires us to find the probability that fewer than seven patients out of a group of twelve recover. The drug is only approved if at least seven out of the twelve patients recover. Therefore, to find the probability that the drug is discredited, we need to find the sum of the probabilities that six or fewer patients recover.

Step 2: Calculation Of Binomial Probability

Knowing from the binomial distribution that the formula to calculate the probability \(P\), for a given number \(k\) of successful trials out of \(n\) trials, with a probability \(p\) of success in a single trial, is given by: \[P(k;n,p) = \binom{n}{k}p^k(1-p)^{n-k}\] , we compute the binomial probability for \(k = 0, 1, 2, ..., 6\), given \(n = 12\) and \(p = 0.5\)

Step 3: Summing Up The Probabilities

To get the total probability that fewer than seven patients recover (which leads to the discreditation of the drug), the individual probabilities calculated in step 2 have to be summed up.

Key Concepts

Binomial Distribution

The binomial distribution is a common probability distribution that models the number of successes in a fixed number of independent experiments, each with the same probability of success. In this context, a "success" could mean a patient recovering after treatment. This distribution helps in understanding how likely it is that a certain number of patients will recover out of a group of 12, assuming each patient has an independent 50% chance of recovery.

For binomial distribution, the main components you need to know are:

• n: The total number of trials or patients, which is 12 in our exercise.
• p: The probability of success on each trial, here it is 0.5 (50%).
• k: The number of successes we would like to find the probability for, like 0, 1, 2, through 6.

These concepts are crucial in applying the binomial distribution formula to calculate specific probabilities.

Probability Calculation

Calculating probabilities using the binomial model involves understanding the formula and applying it rigorously for each possible number of successes. Suppose you wish to calculate the probability of exactly six patients recovering; the formula is:

\[P(k=6; n=12, p=0.5) = \binom{12}{6} (0.5)^6 (0.5)^{12-6}\]

Here’s how to calculate it step-by-step:

• Step 1: Calculate the binomial coefficient \( \binom{n}{k} \) using the formula \( \binom{n}{k} = \frac{n!}{k!(n-k)!} \).
• Step 2: Raise the probability of success \(p\) to the power of \(k\), which is 6.
• Step 3: Raise the probability of failure \((1-p)\) to the power of \(n-k\), which is the number of non-recovering patients.

Repeat these steps for each number from 0 through 6 and add them up to get the total probability of discreditation.

Statistical Hypothesis Testing

Statistical hypothesis testing is a method used to make decisions or infer conclusions about a population based on sample data. Using hypotheses, we test whether a result significantly differs from a specified value under the null hypothesis. In this case, the null hypothesis could state that the new drug is ineffective in improving recovery rates beyond the standard treatment's recovery probability.

To "reject" or "not reject" our null hypothesis, we measure probabilities:

• If the cumulative probability of observing 6 or fewer recoveries is low, then the treatment significantly shows no improvement, and the null hypothesis cannot be rejected.
• If this probability was, however, high (suggesting fewer recoveries are likely by chance), rejecting the null hypothesis would mean we find evidence supporting the new treatment's potential efficacy.

Understanding this helps us decide whether the observed result (fewer recoveries) is due to random chance or a truly ineffective treatment.