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In clinical trials of a medication whose purpose is to reduce the pain associated with migraine headaches, \(2 \%\) of the patients in the study experienced weight gain as a side effect. Suppose a random sample of 600 users of this medication is obtained. Use the normal approximation to the binomial to (a) approximate the probability that exactly 20 will experience weight gain as a side effect. (b) approximate the probability that 20 or fewer will experience weight gain as a side effect. (c) approximate the probability that 22 or more patients will experience weight gain as a side effect. (d) approximate the probability that between 20 and 30 patients, inclusive, will experience weight gain as a side effect.

Step by step solution

01

Define the problem

We need to approximate the binomial probabilities using the normal distribution for a large sample size of 600 with a success probability of 0.02 (weight gain).

02

Calculate the mean and standard deviation

For a binomial distribution, the mean \(\text{μ}\) and standard deviation \(\text{σ}\) are given by:\[ \text{μ} = np = 600 \times 0.02 = 12 \] \[ \text{σ} = \sqrt{np(1-p)} = \sqrt{600 \times 0.02 \times 0.98} ≈ 3.43 \]

03

Approximate the probability for part (a)

To approximate \(P(X=20)\) using the normal distribution, we apply the continuity correction:\[ P(19.5 < X < 20.5) \] Convert to the z-score: \[ z_1 = \frac{19.5 - 12}{3.43} ≈ 2.19 \] \[ z_2 = \frac{20.5 - 12}{3.43} ≈ 2.48 \] Use the standard normal distribution table to find the corresponding probabilities:\[ P(X=20) ≈ P(2.19 < Z < 2.48) = \Phi(2.48) - \Phi(2.19) ≈ 0.4938 - 0.4857 = 0.0081 \]

04

Approximate the probability for part (b)

To approximate \(P(X \leq 20)\), apply the continuity correction:\[ P(X \leq 20.5) \] Convert to the z-score: \[ z = \frac{20.5 - 12}{3.43} ≈ 2.48 \] \[ P(X \leq 20) ≈ \Phi(2.48) = 0.9934 \]

05

Approximate the probability for part (c)

To approximate \(P(X \geq 22)\), apply the continuity correction:\[ P(X \geq 21.5) \] Convert to the z-score: \[ z = \frac{21.5 - 12}{3.43} ≈ 2.77 \] \[ P(X \geq 22) ≈ 1 - \Phi(2.77) = 1 - 0.9973 = 0.0027 \]

06

Approximate the probability for part (d)

To approximate \(P(20 \leq X \leq 30)\), apply the continuity correction:\[ P(19.5 < X < 30.5) \] Convert to the z-score: \[ z_1 = \frac{19.5 - 12}{3.43} ≈ 2.19 \] \[ z_2 = \frac{30.5 - 12}{3.43} ≈ 5.39 \] \[ P(20 \leq X \leq 30) ≈ \Phi(5.39) - \Phi(2.19) = 1 - 0.4857 = 0.5143 \]

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Key Concepts

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

binomial distribution

The binomial distribution is a fundamental concept in probability and statistics. It describes the number of successes in a fixed number of independent trials, where each trial has a binary outcome: success or failure. For example, if you're flipping a coin 10 times, and you want to know the probability of getting exactly 4 heads, you'd use the binomial distribution. In our problem, we're looking at whether patients experience weight gain (success) from a migraine medication trial.

continuity correction

Continuity correction is used to improve the approximation of a discrete distribution (like the binomial) with a continuous one (like the normal). Since the binomial distribution deals with whole numbers, using the normal distribution directly can introduce inaccuracies. To counter this, we adjust the values by 0.5. For example, to approximate the probability of exactly 20 successes, we consider the interval from 19.5 to 20.5.

z-score calculation

The z-score is a measure that describes the position of a value relative to the mean of a group of values. It is calculated as \(z = \frac{X-μ}{σ}\), where \(X\) is the value, \(μ\) is the mean, and \(σ\) is the standard deviation. In our context, the z-score translates the number of patients into a standard metric. For instance, for 20.5 patients (after applying continuity correction), the z-score is calculated by \(z = \frac{20.5 - 12}{3.43}\). This helps us use the standard normal distribution to find probabilities.

probability approximation

Probability approximation involves using the normal distribution to estimate the probabilities of various outcomes in a binomial setting. We start by calculating the mean (\(μ = np\)) and standard deviation (\(σ = \sqrt{np(1-p)}\)). Then, apply continuity correction and convert the range into z-scores. Finally, use the standard normal distribution table (or a calculator) to find the probabilities. For example, for 20 or fewer patients experiencing a side effect, we approximate \(P(X \leq 20)\) using \(P(X \leq 20.5)\), which translates to finding \(Φ(z)\) for the z-score of 20.5.