Problem 11 - A Modern Introduction to Probability and Statistics: Understanding Why and How

Question

One is given a number \(t\), which is the realization of a random variable \(T\) with an \(N(\mu, 1)\) distribution. To test \(H_{0}: \mu=0\) against \(H_{1}: \mu \neq 0\), one uses \(T\) as the test statistic. One decides to reject \(H_{0}\) in favor of \(H_{1}\) if \(|t| \geq 2\). Compute the probability of committing a type I error.

Step-by-Step Solution

Answer

The probability of committing a type I error is 0.0456.

Step 1: Understanding Type I Error

A Type I error occurs when the null hypothesis \(H_0: \mu = 0\) is rejected when it is true. This type of error corresponds to the test statistic falling into the rejection region when the null hypothesis is true.

Step 2: Identifying the Rejection Region

The rejection region for the hypothesis test is defined by \(|t| \geq 2\). This means that when the absolute value of the observed test statistic \(t\) is greater than or equal to 2, we reject the null hypothesis.

Step 3: Normal Distribution Basics

Since the test statistic \(T\) follows an \(N(0,1)\) distribution under the null hypothesis, we need to find the probability that \(|T| \geq 2\).

Step 4: Computing Probability of Rejection

For a standard normal distribution, we compute the probability that \(|T| \geq 2\), which is \(P(T \geq 2) + P(T \leq -2)\). This is equal to \(2 imes P(T \geq 2)\).

Step 5: Using Z-tables to Find Probability

Using a standard normal table or calculator,find \(P(T \geq 2)\). This probability is approximately \(0.0228\). Thus, \(P(|T| \geq 2) = 2 imes 0.0228 = 0.0456\).

Step 6: Conclusion

The probability of committing a type I error, which is the probability of rejecting a true null hypothesis, is approximately \(0.0456\).

Key Concepts

Type I Error

A Type I error is one of the possible errors that can occur during hypothesis testing. It is made when the null hypothesis \( H_0 \) is incorrectly rejected, even though it is actually true. This means you conclude that there is an effect, difference, or relationship when in reality there isn't one.

In our specific exercise, the null hypothesis \( H_0: \mu = 0 \) means we assume there's no difference or no effect unless evidence suggests otherwise. A Type I error would happen if we reject this assumption mistakenly, thinking \( \mu eq 0 \) when it is indeed 0.

The importance of knowing and controlling the Type I error lies in its impact on research conclusions. By setting a significance level, often denoted as \( \alpha \), we control the probability of making this error. In our case, the probability of committing a Type I error was calculated as approximately 0.0456, indicating a 4.56% chance of mistakenly rejecting a true null hypothesis.

Normal Distribution

In hypothesis testing, the concept of a normal distribution forms the backbone of many statistical procedures. Normal distribution, often referred to as a Gaussian distribution, is characterized by its bell-shaped curve which is symmetric around the mean.

In the exercise, our test statistic \( T \) follows a normal distribution \( N(0,1) \) under the null hypothesis \( H_0: \mu = 0 \). This special case is called a standard normal distribution where the mean \( \mu \) is 0 and the standard deviation \( \sigma \) is 1.

Normal distribution is incredibly important because it allows statisticians to use z-tables or standard normal distribution calculators to determine probabilities related to test statistics. It gives us the ability to understand the likelihood of observing a test statistic as extreme as \(|t| \geq 2\), thus aiding in decision-making during hypothesis testing.

Rejection Region

The rejection region is a crucial concept in hypothesis testing. This is the set of values for the test statistic which leads to the rejection of the null hypothesis \( H_0 \). It is often determined by a pre-set significance level, \( \alpha \), which corresponds to the highest probability of committing a Type I error you are willing to accept.

In our exercise, the rejection region is defined as \(|t| \geq 2\). This means that if the absolute value of the observed test statistic is greater than or equal to 2, we have enough evidence to reject the null hypothesis \( H_0: \mu = 0 \). The boundary of this region takes into account both extremes of the normal distribution: on both sides, positive \( t \) and negative \( t \).

By identifying and understanding the rejection region, researchers can determine which observed values suggest a significant effect or difference from what is stated in the null hypothesis. This is key to making informed conclusions based on statistical data.