Question

Let \(x\) have a binomial distribution with \(n=50\) and \(\pi=.6\), so that \(\mu=n \pi=30\) and \(\sigma=\sqrt{n \pi(1-\pi)}=\) 3.4641. Calculate the following probabilities using the normal approximation with the continuity correction: a. \(P(x=30)\) b. \(P(x=25)\) c. \(P(x \leq 25)\) d. \(P(25 \leq x \leq 40)\) e. \(P(25

Short answer

Probabilities will be calculated using the z-scores and the area under the standard Normal curve. Remember to use the continuity correction for each x value, and the probabilities for each case will depend on the specific z scores you generate.

Step-by-step solution

Calculating Probability using Normal Approximation

For calculating these probabilities using a normal approximation with continuity correction, standardize each value of x by subtracting the mean \(\mu\) and dividing by the standard deviation \(\sigma=\sqrt{n \pi(1-\pi)}\). This will yield z scores.

Applying Continuity Correction and Using Standard Normal Table

The continuity correction requires that you add or subtract 0.5 to the x values depending on the inequality in the exercise. This is crucial to approximate the discrete binomial distribution to the continuous normal distribution. Then you will look up the z values in the standard normal table (or use an online calculator) to find the corresponding probabilities.

Step by Step Probabilities Calculation

Calculate each of the given quantities by applying the above principles. For example: a. \(P(x=30)\) approximates to \(P(29.5 \leq x \leq 30.5)\), b. \(P(x=25)\) approximates to \(P(24.5 \leq x \leq 25.5)\), and so on. Just ensure to subtract \(\mu\) and divide by \(\sigma\) for each x in the range and find the corresponding z score.

Key concepts

Binomial Distribution

The binomial distribution is a discrete probability distribution that describes the number of successes in a fixed number of independent trials, each with the same probability of success. In our exercise, the number of trials, or n, is 50, and the probability of success, or \(\rho\), is 0.6 for each trial. The mean, \(\mu\), is given by n\(\rho\), and the standard deviation, \(\sigma\), by \(\sqrt{n \rho(1-\rho)}\). When we calculate probabilities using the binomial distribution, we're essentially asking how likely it is for a certain number of successes to occur in that set of trials.

Standard Normal Distribution

The standard normal distribution is a continuous probability distribution that is symmetric around the mean, which is zero, and has a standard deviation of one. This distribution is famous for the 68-95-99.7 rule, which signifies that approximately 68% of the data falls within one standard deviation of the mean, 95% within two, and 99.7% within three. In the context of our exercise, we are using the properties of the standard normal distribution to approximate probabilities from the binomial distribution. To do this, we convert the binomial distribution to a standard normal distribution through standardization, which is where the z-score comes into play.

The conversion to z-scores allows us to use the standard normal table, a tool which lists the probability that a standard normal random variable is less than or equal to a given value.

Continuity Correction

Since the binomial distribution is discrete and the normal distribution is continuous, we need a method to bridge the gap between them when we approximate binomial probabilities using the normal distribution. This is where continuity correction comes into play. It is a technique used to improve the approximation by adjusting the discrete x values to continuous intervals. For example, when calculating P(x=30), we adjust to P(29.5 \(\<=\) x \(\<=\) 30.5) to account for the entire interval that the single value 30 would cover if the distribution were continuous.

By applying the continuity correction, we ensure the discrete nature of binomial distribution is more accurately represented when using the normal model. This simple adjustment is crucial to minimize the discrepancy between the discrete and continuous probabilities.

Z Score Analysis

A z score analysis involves finding the number of standard deviations a data point is from the mean. In standardized form, the z score indicates the position of the original value in the standard normal distribution. Calculating the z score requires subtracting the mean from the data value and dividing the result by the standard deviation. For the binomial distribution, this allows us to convert our discrete values to a form that's compatible with the continuous standard normal distribution.

In calculating probabilities like P(25 \(\<=\) x \(\<=\) 40), we first apply continuity correction and then find the respective z scores for the adjusted range. These z scores allow us to use the standard normal table to find the corresponding probabilities, thus completing our approximation of the binoidal probabilities with the aid of the standard normal distribution.