Question

In a psychology experiment, the researcher designs a maze in which a mouse must choose one of two paths, colored either red or blue, at each of 10 intersections. At the end of the maze, the mouse is given a food reward. The researcher counts the number of times the mouse chooses the red path. If you were the researcher, how would you use this count to decide whether the mouse has any preference for color?

Short answer

Answer: We can determine if a mouse has a preference for a specific color by calculating the expected number of red path choices assuming no preference, finding probabilities of extreme outcomes using the binomial probability formula, summarizing the probabilities, and comparing the actual number of red choices to those probabilities. If the actual count falls within the range of low probabilities, it may indicate a preference. If it is close to the expected value, there is no strong evidence of preference.

Step-by-step solution

Determine the Expected Number of Red Path Choices

If the mouse had no preference for either color, it would choose the red or blue path with equal probability. Hence, at any intersection, there is a 1/2 chance it selects the red path, and a 1/2 chance it selects the blue path. With 10 intersections, the expected number of times that the mouse would choose the red path without any preference would be: Expected number = (number of intersections) * (probability of choosing red) E = 10 * (1/2) = 5

Calculate the Probabilities of Extreme Outcomes

To determine if the mouse has a preference for either color, we need to calculate the probability of observing a number of red choices greater or less than the expected outcome. This can be done using the binomial probability formula: P(X = k) = (n choose k) * p^k * (1-p)^(n-k) In this problem, "n" is the number of intersections (10), "p" is the probability of choosing the red path (1/2), and "k" is the number of red choices. We need to calculate probabilities for k below 5 or above 5 since extreme outcomes in either direction could suggest the mouse has a preference. For example, if the mouse chooses the red path all ten times or never, that would be an extreme outcome.

Summarize the Probabilities

Calculate the probabilities of all possible outcomes for k, from 0 to 10. After obtaining the probability values, look at the sum of all probabilities below and above the expected value of 5. For instance, if the sum of probabilities for k = 0 to k = 4 is very low, it would indicate that the mouse's choice of the red path is significantly less frequent than expected by chance. Similarly, if the sum of probabilities for k = 6 to k = 10 is very low, it means the mouse is choosing the red path significantly more often than expected by chance.

Determine the Presence of Preference

Based on the probabilities obtained in step 3, decide whether the mouse has any preference for color. If the actual number of red choices falls within the range of low probabilities, that could be evidence of a preference. On the other hand, if the actual count is close to the expected value of 5, there is no strong evidence of preference for either color. In conclusion, using the expected number of red path choices, calculating the probabilities of more extreme outcomes, and by comparing the actual number of red choices to those probabilities, we can make an informed decision about whether the mouse has any preference for the red or blue path.

Key concepts

Expected Value

The expected value is an important concept in probability theory that provides a measure of the central tendency of a random variable. In simpler terms, it tells us what average outcome we can expect if an experiment is repeated many times. In this maze experiment, if the mouse were making random choices at each intersection, we would expect it to choose the red path half the time.
To calculate this, we multiply the number of paths (10 intersections) by the probability of choosing the red path, which is 1/2. This gives us an expected number of red path choices:

• Expected number of red choices = 10 * (1/2) = 5

This means, on average, the mouse is expected to choose the red path 5 times if it has no particular preference. Understanding the expected value is crucial before diving into further analysis of the mouse's path choices.

Probability Distribution

Probability distribution describes how the probabilities are distributed over the different possible outcomes of a random experiment. In the context of our experiment with the mouse, we assume a binomial probability distribution. This is because we have a fixed number of trials (10 intersections) and two possible outcomes for each trial (red or blue path).
To determine if the mouse has a path color preference, we look at the likelihood of various choices using the binomial distribution formula:

• \[P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}\]
• "n" is the number of trials, "p" is the probability of choosing red, and "k" is the number of successful trials (red path choices).

By calculating the probabilities for all possible numbers of red choices (k = 0 to 10), we form a complete probability distribution. Extreme values (far from 5) could suggest non-random behavior, possibly indicating a preference for red or blue.

Hypothesis Testing

Hypothesis testing is a statistical method used to decide whether there is enough evidence to support a specific hypothesis. In our experiment, the hypothesis is that the mouse has no preference between red and blue. When we conduct hypothesis testing here, we assess this through the probability distribution we discussed earlier.
First, identify an expected outcome and its nearby extreme outcomes:

• If the actual number of red choices is significantly different from 5, it might suggest a preference.
• We calculate the probability of these extreme outcomes to statistically determine the presence of any bias.

If the actual number of red path choices happens to be in the region where there's a low probability, we may reject the null hypothesis (no preference). Thus, hypothesis testing helps to conclude whether the mouse displays a statistically significant preference based on the observed behavior compared with the expected randomness.