Problem 5 - An Introduction to Mathematical Statistics and Its Applications

Question

Suppose that \(n=7\) paired observations, \(\left(X_{i}, Y_{i}\right)\), are recorded, \(i=1,2, \ldots, 7\). Let \(p=P\left(Y_{i}>X_{i}\right)\). Write out the entire probability distribution for \(Y_{+}\), the number of positive differences among the set of \(Y_{i}-X_{i}\) 's, \(i=1,2, \ldots, 7\), assuming that \(p=\frac{1}{2} .\) What \(\alpha\) levels are possible for testing \(H_{0}: p=\frac{1}{2}\) versus \(H_{1}: p>\frac{1}{2}\) ?

Step-by-Step Solution

Answer

The entire probability distribution for \(Y_{+}\) is given by \(P(Y=k)=C(7,k)*1/2^7\) where \(k = 0, 1, 2, ..., 7\). The possible \(\alpha\) levels for testing \(H_0: p = 0.5\) versus \(H_1: p > 0.5\) will be given by the probability \(P(Y \geq c)\) under the null hypothesis, for each cutoff value \(c=0, 1, 2, ..., 7\). The computation for it would need the given binomial distribution.

Step 1: Define the Binomial Distribution

A binomial distribution is defined as \(Bin(n, p)\) where \(n\) is the number of observations/experiments and \(p\) is the probability of success in each observation. Here, the success is that \(Y_{i}>X_{i}\) for a pair of readings. As per the problem, \(n=7\) and \(p=0.5\).

Step 2: Write the Probability Distribution

The probability distribution of a binomial variable \(Y\) is given by: \(P(Y=k) = C(n,k)\(p^k) * \((1-p)^(n-k)\) where \(C(n,k) = n! / \((k! * (n-k)!)\) with \(k\) ranging from 0 to \(n\). Substituting \(n=7\) and \(p=0.5\) into the binomial distribution, we get \(P(Y=k)\) equals to \(C(7, k) * (0.5)^k * ((1-0.5)^(7-k)) = C(7, k) * 1/2^7\), \(k = 0, 1, 2, ..., 7\).

Step 3: Calculate the Possible Alpha Levels

In order to test the hypothesis \(H_0 : p = 0.5\) versus \(H_1 : p > 0.5\), we need to consider possible rejection regions, say if \(Y >= c\) for some particular \(c\) value (0 to 7). The p-value or alpha levels is the probability that we observe a statistic as extreme as the one observed, under the assumption that the null hypothesis is true. Therefore, given the distribution under \(H_0\), the alpha level for a particular \(c\) will be the probability \(P(Y >= c)\), when \(p = 0.5\), computed using the binomial distribution calculated in Step 2.

Key Concepts

Probability Distribution

Imagine you're comparing observations to see how often one event happens over another. In our exercise, we're interested in how many times the value of \(Y_i\) is greater than \(X_i\) across 7 pairs. We use a **binomial distribution** to model this scenario.
This distribution is perfect for cases where you have a fixed number of trials, each with two possible outcomes, like 'success' or 'failure'. Here, a 'success' is when \(Y_i > X_i\).

• **Trials (n):** 7 observations
• **Probability of Success (p):** \(p=0.5\) because each outcome is equally likely

The probability of getting exactly \(k\) successes (where \(0 \leq k \leq 7\)) is given by the formula:\[P(Y=k) = \binom{7}{k} \left(0.5\right)^k \left(0.5\right)^{7-k}\]Understanding this helps you write out the full probability distribution, which lists the probability for each possible number of successes from 0 to 7.

Hypothesis Testing

Hypothesis testing is a way to test claims or ideas about a data sample. In our exercise, you have a **null hypothesis \(H_0\)** that assumes the probability \(p\) is 0.5. The **alternative hypothesis \(H_1\)** is \(p > 0.5\), suggesting that \(Y_i\) is greater than \(X_i\) more often than not.
This testing is crucial for deciding whether the observed data can occur under the null hypothesis. You're looking for evidence strong enough to reject \(H_0\) in favor of \(H_1\).

• **Null Hypothesis (\(H_0\)):** \(p = 0.5\)
• **Alternative Hypothesis (\(H_1\)):** \(p > 0.5\)

Through testing, you assess whether your data significantly deviate from the null hypothesis or not, thus either supporting or refuting \(H_0\).

Significance Level

The **significance level**, often denoted by \(\alpha\), is a threshold you set to decide whether an observed effect is statistically significant. It represents the risk of rejecting the null hypothesis when it is actually true.
In the context of our problem, the significance level is the probability of observing a result as extreme as the actual observed result, assuming the null hypothesis is true.
To find appropriate \(\alpha\) levels, you consider values of \(k\) where \(P(Y \geq k)\) provides strong evidence against \(H_0\). This means finding \(k\) such that if \(Y\) is at least \(k\), we significantly doubt \(H_0\):

• For example, choosing \(k\) where \(\alpha = P(Y \geq k)\) ensures only results with enough evidence lead you to reject \(H_0\).

Choosing the right \(\alpha\) level helps balance the risks of making type I errors (false positives) and aids in credible decision-making during hypothesis testing.