Question

Consider a binomial experiment with \(n=20\) and \(p=.4 .\) Calculate \(P(x \geq 10)\) using each of these methods: a. Table 1 in Appendix I b. The normal approximation to the binomial probability distribution

Short answer

Answer: Using the binomial probability table, calculate the cumulative probability for x-values from 10 to 20. For the normal approximation, find the z-score and use a z-table to determine the probability. The probability of getting at least 10 successes is approximately 0.2514 using the normal approximation method.

Step-by-step solution

Locate the relevant row and column in Table 1

Table 1 provides the binomial probability values for given n and p values. Locate the row corresponding to n = 20 and the column for p = 0.4.

Find the probabilities for each x-value

Since we need to find the probability for x >= 10, we need to find the probabilities for each x-value from 10 to 20. Find the probabilities for all these x-values given by the table.

Calculate the total probability

Add the probabilities of x-values from 10 to 20 to find the total cumulative probability for x >= 10. Method b: Normal approximation to the binomial probability distribution

Calculate the mean and standard deviation

To use the normal approximation, we first need the mean and standard deviation of the binomial distribution. Calculate the mean and standard deviation using the given values of n and p: Mean (\(\mu\)) = np = 20 * 0.4 = 8 Standard deviation (\(\sigma\)) = \(\sqrt{np(1-p)}\) = \(\sqrt{20*0.4*0.6}\) = 2.19

Apply the Continuity Correction Factor (CCF)

Since we want to find the probability P(x ≥ 10), we will apply the Continuity Correction Factor (CCF) to the value of x, which is x = 9.5. This corrects for the fact that we are approximating a discrete distribution with a continuous one.

Calculate the z-score

A z-score helps determine how many standard deviations an element is from the mean. Calculate the z-score for x = 9.5 using the formula: z = (x - μ) / σ = (9.5 - 8) / 2.19 = 0.68

Use a z-table to find the probability

Look up the probability for z = 0.68 from a standard normal distribution table, which is 0.7486.

Calculate P(x ≥ 10) using the z-table result

The value 0.7486 corresponds to P(x ≤ 9.5), but we need the probability for x ≥ 10. To get this, we need to subtract the result from 1: P(x ≥ 10) = 1 - P(x ≤ 9.5) = 1 - 0.7486 = 0.2514 So the probability P(x ≥ 10) using the normal approximation is approximately 0.2514.

Key concepts

Normal Approximation

In probability theory, the normal approximation is a convenient method to estimate binomial probabilities, especially when dealing with large sample sizes. A binomial distribution, which is based on discrete events, can be approximated by a normal distribution when certain conditions are met:

• The number of trials \(n\) is large enough.
• The probability of success \(p\) is neither very close to 0 nor 1.

The normal approximation applies the bell-shaped curve of the normal distribution to represent the discrete distribution. This curve represents continuous data, simplifying complex calculations associated with binomial probabilities. However, it is essential to note that the continuity correction factor (explained later) must be applied, as this method shifts from discrete to continuous distributions.

Probability Calculation

Calculating probabilities using the normal approximation involves several critical steps.First, identify the mean \((\mu)\) and standard deviation \((\sigma)\) of the binomial distribution:

• Mean \((\mu)\) is calculated as \(np\).
• Standard deviation \((\sigma)\) is determined by \(\sqrt{np(1-p)}\).

These calculations provide the foundation for approximating the distribution to a normal curve.Next, to find the specific probability, such as \(P(x \geq 10)\), one must compute the z-score. The z-score indicates how many standard deviations a particular value \(x\) is from the mean. The formula is:\[z = \frac{x - \mu}{\sigma}\]Once the z-score is calculated, use a z-table to find the corresponding probability value. For example, if the z-score is 0.68, you can find that the probability for \(P(x \leq 9.5)\) is 0.7486. To convert this to \(P(x \geq 10)\), you subtract the value from 1.

Continuity Correction Factor

The continuity correction factor is a crucial step when using normal approximation for a binomial distribution. Since a binomial distribution is discrete, values are distinct whole numbers. In contrast, a normal distribution is continuous. When we switch from one to the other in approximation, a small adjustment is necessary. This is where the continuity correction factor comes in.When calculating \(P(x \geq 10)\), you actually consider \(x \geq 9.5\). This adjustment accounts for the entire interval that \(x\) covers on a continuous scale. By applying the continuity correction factor, you ensure that your probability estimation more accurately captures the behavior of the original binomial distribution. Including this step is essential when transitioning from discrete numbers to continuous probabilities, which ultimately enhances the accuracy of your results.