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Chapter 9: Problem 74

A Maze Experiment In a maze running study, a rat is run in a T maze and the result of each run recorded. A reward in the form of food is always placed at the right exit. If learning is taking place, the rat will choose the right exit more often than the left. If no learning is taking place, the rat should randomly choose either exit. Suppose that the rat is given \(n=100\) runs in the maze and that he chooses the right exit \(x=64\) times. Would you conclude that learning is taking place? Use the \(p\) -value approach, and make a decision based on this \(p\) -value.

Short Answer

Expert verified
Answer: To answer this question, follow the steps in the provided solution and calculate the p-value. If the p-value is less than or equal to 0.05, we can conclude that learning is taking place; otherwise, we cannot determine if learning is taking place based on the given data.

Step by step solution

01

Define the null hypothesis and alternative hypothesis

The null hypothesis (\(H_0\)) states that there is no learning, meaning the rat picks either exit with equal probability (50% chance for each). The alternative hypothesis (\(H_A\)) states that the rat is learning and chooses the right exit more often than the left. \(H_0: p = 0.5\) \(H_A: p > 0.5\)
02

Calculate the test statistic

Use the formula for the test statistic, Z, for a proportion test: \(Z= \frac{\bar{p} - p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}}\) where \(\bar{p}\) is the proportion of right exits chosen in the experiment, \(p_0\) is the proportion of right exits under the null hypothesis, and \(n\) is the number of trials. In this case, \(\bar{p} = \frac{64}{100} = 0.64\), \(p_0 = 0.5\), and \(n=100\). \(Z = \frac{0.64 - 0.5}{\sqrt{\frac{0.5(1-0.5)}{100}}}\) Calculate the Z value.
03

Calculate the p-value

Using a standard normal distribution table or calculator, find the probability that the test statistic (Z-value) is greater than or equal to the calculated test statistic. \(p-value = P(Z 鈮� Z_{calculated})\) For our case, find the p-value for the calculated Z value.
04

Compare the p-value to the significance level

Compare the calculated p-value to a predetermined significance level (\(\alpha\)). A common choice is \(\alpha = 0.05\). If the p-value is less than or equal to \(\alpha\), then reject the null hypothesis (\(H_0\)) and conclude that learning is taking place. If the p-value is greater than \(\alpha\), then fail to reject the null hypothesis (\(H_0\)) and conclude that it is not possible to determine if learning is taking place based on the given data.
05

Interpret the result

Based on the calculated p-value and comparison to the significance level, make a conclusion about whether learning is taking place in the maze experiment.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Proportion Test
Proportion tests help us determine if there's a significant difference between a sample proportion and a hypothesized population proportion. In the maze experiment, we want to see if the rat learned to consistently choose the right exit. Here, we utilize a one-sample proportion test, which is appropriate for this type of probability question.

To conduct a proportion test, you:
  • Define the null hypothesis (\(H_0\)) that there is no effect or difference, meaning the rat picks either exit 50% of the time.
  • Define the alternative hypothesis (\(H_A\)) that suggests a difference, meaning the rat picks the right exit more than 50% of the time.
This forms the foundation for assessing whether we can conclude if learning has occurred.
P-Value Approach
The p-value approach is essentially about measuring the strength of the evidence against the null hypothesis. In this method, the p-value is computed from the test statistic to determine how extreme the observed result is, under the assumption that the null hypothesis is true.

  • A small p-value (typically 鈮� 0.05) indicates strong evidence against the null hypothesis, allowing us to reject it.
  • A large p-value indicates weak evidence against the null hypothesis, and we fail to reject it.
In the maze experiment, once we calculate the Z-value using the formula for the proportion test, the p-value is determined. This p-value reflects the likelihood of observing our data given the assumption that the rat is not learning (i.e., chooses either exit 50% of the time).
Alternative Hypothesis
The alternative hypothesis is a critical part of hypothesis testing, representing what we aim to demonstrate or what we suspect might be true. In the context of the maze experiment:
  • The null hypothesis (\(H_0\)) is that the rat shows no learning鈥攃hoosing either the left or right exit equally.
  • The alternative hypothesis (\(H_A\)) is that the rat is learning鈥攃hoosing the right exit more frequently than the left.
By focusing on the alternative hypothesis, we assess whether there is evidence to support a significant difference from what we would expect under random chance. This sets the stage for testing and potentially rejecting the null hypothesis.
Significance Level
The significance level, often denoted by alpha (\(\alpha\)), is a threshold set by the researcher to determine when to reject the null hypothesis. In many studies, a common significance level of 0.05 is chosen.
  • If the p-value is less than or equal to the significance level, it suggests significant evidence against the null hypothesis, prompting its rejection.
  • If the p-value is greater than the significance level, there is not enough evidence to reject the null hypothesis, and it is retained.
In our maze experiment, we might choose a significance level of 0.05. If the calculated p-value is below this level, it indicates sufficient evidence that the rat is indeed learning its way through the maze more often, by consistently taking the right exit.

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Most popular questions from this chapter

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