"Blind Emotion [See Problem 11 in Section 10.2A.] When the area of the brain responsible for vision is destroyed, individuals experience cortical blindness. Patients with cortical blindness are unaware of any visual stimulus including light. In a 52-year-old male patient with cortical blindness (as a result of two strokes within a 38-day timeframe), a series of visual stimuli were presented on a computer screen. The patient was given two choices for each stimulus and asked to report what was on the screen. The patient’s responses were recorded by an individual who could not see the contents on the screen.
a. An initial baseline test was conducted by presenting black squares and circles on the white background of the screen. The patient correctly “guessed” the item on the screen in 90 of the 200 trials. Researchers concluded the patient’s performance was not statistically different from chance. Explain what this means using the language of hypothesis testing.
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Verified step by step guidance

1. Step 1: Identify the null hypothesis (H_0) and the alternative hypothesis (H_a). In this context, the null hypothesis states that the patient is guessing randomly, so the probability of a correct guess is 0.5 (chance level). The alternative hypothesis states that the patient’s performance is different from chance, meaning the probability of a correct guess is not 0.5.
2. Step 2: Define the test statistic. Since the patient’s responses are binary (correct or incorrect), and the number of trials is large, we can model the number of correct guesses as a binomial random variable. The test statistic can be approximated by a normal distribution using the formula for the z-score: \[ \text{z} = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}} \] where \(\hat{p}\) is the observed proportion of correct guesses, \(p_0\) is the hypothesized probability under the null (0.5), and \(n\) is the number of trials.
3. Step 3: Calculate the observed proportion of correct guesses: \[ \hat{p} = \frac{90}{200} \] This represents the patient’s empirical success rate.
4. Step 4: Compute the z-score using the formula from Step 2 by substituting \(\hat{p}\), \(p_0 = 0.5\), and \(n = 200\). This z-score measures how far the observed proportion is from the chance level in terms of standard errors.
5. Step 5: Interpret the result by comparing the z-score to critical values from the standard normal distribution for a chosen significance level (e.g., 0.05). If the z-score falls within the acceptance region (not extreme enough), we fail to reject the null hypothesis, meaning the patient’s performance is not statistically different from chance. This implies the observed correct guesses could reasonably occur by random guessing.

Key Concepts

Hypothesis Testing

Hypothesis testing is a statistical method used to decide whether there is enough evidence to reject a null hypothesis. It involves comparing observed data to what we would expect under the null hypothesis, often using a significance level to determine if results are due to chance or indicate a real effect.

Null Hypothesis and Alternative Hypothesis

The null hypothesis typically states that there is no effect or difference, such as the patient guessing correctly by chance. The alternative hypothesis suggests there is an effect, like the patient performing better than chance. Testing evaluates whether observed results provide sufficient evidence to reject the null.

Statistical Significance and Chance Performance

Statistical significance indicates whether observed results are unlikely to have occurred by chance alone. In this context, saying the patient’s performance is not statistically different from chance means the number of correct guesses could be explained by random guessing, so no evidence supports the patient perceiving the stimuli.