Question

A psychiatrist believes that \(80 \%\) of all people who visit doctors have problems of a psychosomatic nature. She decides to select 25 patients at random to test her theory.a. Assuming that the psychiatrist's theory is true, what is the expected value of \(x\), the number of the 25 patients who have psychosomatic problems? b. What is the variance of \(x\), assuming that the theory is true? c. Find \(P(x \leq 14)\). (Use tables and assume that the theory is true.) d. Based on the probability in part \(c\), if only 14 of the 25 sampled had psychosomatic problems, what conclusions would you make about the psychiatrist's theory? Explain.

Short answer

Answer: The expected number of patients with psychosomatic problems is 20, the variance is 4, and the probability that 14 or fewer patients have psychosomatic problems is approximately 0.99%.

Step-by-step solution

Identify the binomial distribution parameters

In this problem, we have a binomial distribution since we have a fixed number of trials (25 patients), each trial has two possible outcomes (psychosomatic problem or not), the probability of success (psychosomatic problem) is constant, and the trials are independent. We are given the probability of success, \(p\), as 0.8 or 80%. The number of trials, \(n\), is 25.

Calculate the expected value of x

For a binomial distribution, the expected value (mean) is given by the formula: \(E(x) = np\) Using the given values, we have: \(E(x) = 25 \times 0.8 = 20\) Therefore, the expected value of \(x\), the number of patients with psychosomatic problems, is 20.

Calculate the variance of x

For a binomial distribution, the variance is given by the formula: \(Var(x) = np(1-p)\) Using the given values, we have: \(Var(x) = 25 \times 0.8 \times (1 - 0.8) = 25 \times 0.8 \times 0.2 = 4\) So the variance of \(x\) is 4.

Calculate P(x ≤ 14)

To find the probability that the number of psychosomatic patients is less than or equal to 14, we will use the cumulative binomial probability formula, or alternatively, we can use a binomial probability table or calculator. In this case, we will use a table for convenience. For \(n = 25\) and \(p = 0.8\) (80%), we look for \(P(x \leq 14)\) in the table: \(P(x \leq 14) \approx 0.0099\) Thus, the probability that there are 14 or less patients with psychosomatic problems is approximately 0.0099 or 0.99%.

Evaluate the psychiatrist's theory

The low probability (\(0.99 \%\)) of having only 14 or fewer patients with psychosomatic problems, out of the 25 sampled, indicates that this outcome is quite unlikely if the psychiatrist's theory is true. If only 14 of the 25 patients in the sample had psychosomatic problems, which is significantly lower than the expected value of 20, this might lead us to doubt the psychiatrist's theory that 80% of the patients have psychosomatic problems. However, we must keep in mind that this is just a single sample, and the outcome of a single sample may not represent the overall population. To strengthen our conclusions, further experimentation and analysis of more samples would be helpful.

Key concepts

Expected Value

The expected value is a central concept in probability and statistics, particularly when dealing with binomial distributions. It is essentially a measure of the center or average of a random variable. For a binomial distribution, the expected value indicates the average outcome you'd expect from a series of trials, under the assumption that each trial is independent and has the same probability of success. The formula for the expected value in a binomial distribution is given by:

• \( E(x) = np \)

where \( n \) is the number of trials and \( p \) is the probability of success in each trial. In the context of our exercise, using the values \( n = 25 \) and \( p = 0.8 \), the expected value is calculated as:

• \( E(x) = 25 \times 0.8 = 20 \)

This means that, out of 25 patients, we would expect 20 to have psychosomatic problems, were the psychiatrist's theory true.

Variance

Variance measures the degree of spread or dispersion in a set of values. In other words, it tells us how much the values of a random variable deviate from the expected value. For a binomial distribution, the formula is:

• \( Var(x) = np(1-p) \)

Here, \( n \) is the number of trials, \( p \) is the probability of success, and \( (1-p) \) is the probability of failure. In our example, with \( n = 25 \) and \( p = 0.8 \), the variance is calculated as:

• \( Var(x) = 25 \times 0.8 \times (1-0.8) = 25 \times 0.8 \times 0.2 = 4 \)

This variance value of 4 indicates how much the actual number of patients with psychosomatic problems could vary around the expected value of 20. A smaller variance implies that outcomes are closely clustered around the expected value, while a larger variance indicates more spread.

Cumulative Probability

Cumulative probability is the probability that a random variable takes on a value less than or equal to a specific value. In the context of a binomial distribution, it represents the sum of probabilities for all outcomes up to a certain number of successes. Finding the cumulative probability for a binomial distribution often involves using cumulative probability tables or computational tools since calculating it manually can be complex. In our exercise, we're interested in finding the cumulative probability that 14 or fewer of the 25 patients have psychosomatic problems, assuming the psychiatrist's theory is true.With \( n = 25 \) and \( p = 0.8 \), the cumulative probability \( P(x \leq 14) \) is approximately 0.0099 or 0.99%. This low probability suggests that observing 14 or fewer such patients is an uncommon event if the theory holds.

Statistical Hypothesis Testing

Statistical hypothesis testing involves making inferences or decisions about a population based on sample data. It often requires setting up a null hypothesis, which represents a default position (e.g., the theory is true), and an alternative hypothesis, which is what you might accept if evidence suggests the null hypothesis might not hold. In this exercise, the null hypothesis states that 80% of all people visiting doctors have psychosomatic problems. The probability of observing 14 or fewer patients with such problems, given this theory, was calculated at only 0.99%. This small probability suggests that either a rare event has occurred, or the psychiatrist's theory might not be accurate. Yet, a single sample size is just one piece of evidence. Decisions should consider:

• Sample size and representativeness
• Probability threshold for statistical significance
• Potential for bias in sample collection

Continued research and larger or repeated samples could add robustness to our conclusions. It's important to keep a critical mindset, acknowledging the limitations of small sample analyses.