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Chapter 6: Problem 71

Find the following binomial probabilities using the normal approximation. a. \(n=70, \quad p=.30, \quad P(x=18)\) b. \(n=200, \quad p=.70, \quad P(133 \leq x \leq 145)\) c. \(n=85, \quad p=.40, \quad P(x \geq 30)\) d. \(n=150, \quad p=.38, \quad P(x \leq 62)\)

Short Answer

Expert verified
The probability for each case (rounded to 4 decimal places) is: a. P(x=18) = 0.0279, b. P(133 鈮� x 鈮� 145) = 0.8193, c. P(x 鈮� 30) = 0.9333, d. P(x 鈮� 62) = 0.7483.

Step by step solution

01

Convert Binomial to Normal Distribution

To convert a binomial distribution to a normal distribution, the formula \( N(np, \sqrt{npq}) \) will be used, where \( np \) is the mean and \( \sqrt{npq} \) is the standard deviation. Calculate this for each given case a-d: (n=70, p=.30, q=1-.30=.70), (n=200, p=.70, q=1-.70=.30), (n=85, p=.40, q=1-.40=.60), (n=150, p=.38, q=1-.38=.62).
02

Calculate Z-scores

Now, the z-score will be calculated for the x values from each case using the formula \( z = \frac{x - np}{\sqrt{npq}} \). After calculating the z-scores, these values will be matched with the corresponding closest integers that are less than these z-scores.
03

Determine Normal Probabilities

For each case, the corresponding normal probabilities should be determined by looking up each calculated z-score in the standard normal (Z) table (or using a calculator that has the functionality). The result will give the probability for each case. Note that in some cases the probabilities need to be subtracted from 1 if the question asks for more than a particular value. For example in subquestion c, you need to subtract the calculated probability from 1.

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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 statistical method used to model the number of successes in a fixed number of independent experiments, each of which has the same probability of success. In simpler terms, you can think of it like flipping a coin a certain number of times and counting how many times it lands on heads.

Key characteristics of binomial distribution include:
  • It requires a fixed number of trials, denoted by \( n \).
  • Each trial has two possible outcomes, often termed success or failure.
  • The probability of success, denoted by \( p \), remains constant for each trial.
  • The probability of failure is \( q = 1 - p \).
  • The outcomes of all trials are independent.
Understanding these elements is crucial when approaching problems that involve the binomial distribution, like the ones in the exercise. For instance, knowing \( n \) and \( p \) allows you to define the framework needed to solve using the normal approximation when direct binomial calculations are cumbersome.
Normal Distribution
The normal distribution, also known as the Gaussian distribution, is one of the most important continuous probability distributions in statistics. It describes how the values of a variable are distributed. It is symmetrically shaped and often referred to as a "bell curve."

Key properties of the normal distribution include:
  • It is defined by two parameters: the mean (\( \mu \)) and the standard deviation (\( \sigma \)).
  • It is symmetrical around the mean, meaning the left and right sides of the curve are mirror images.
  • 68% of the data falls within one standard deviation of the mean, 95% within two, and 99.7% within three (empirical rule).
When approximating a binomial distribution with a normal distribution, as seen in the exercise, the mean is \( np \) and the standard deviation is \( \sqrt{npq} \). This transformation allows for easier calculation of probabilities when the number of trials is large.
Z-Scores
A z-score measures how many standard deviations an element is from the mean. In other words, it's a way to compare data points coming from different distributions by placing them on the same standard scale.

Calculating a z-score involves:
  • Subtracting the mean from the data point of interest.
  • Dividing the result by the standard deviation.
The formula for calculating a z-score \( z \) is given by: \[ z = \frac{x - \mu}{\sigma} \]In the context of the exercise, \( \mu \) is \( np \) and \( \sigma \) is \( \sqrt{npq} \). Each z-score reveals how far from the mean (in standard deviation terms) a particular x-value is. These values are key to finding probabilities associated with the normal distribution of a variable.
Probability Calculation
Once the z-scores have been calculated, they can be used to find probabilities associated with a normal distribution. This involves using standard normal distribution tables or technology that computes these probabilities.

Steps to calculate probabilities:
  • Calculate the z-score for the value of interest using the aforementioned formula.
  • Use a standard normal distribution table, or a calculator, to find the probability corresponding to the z-score.
  • Remember to apply continuity correction if needed (e.g., subtract a small amount when needing the exact number in a range).
  • If looking for probabilities greater than a given value, subtract the cumulative probability from 1.
The successful conversion of the binomial distribution to a normal distribution simplifies these calculations significantly, especially when dealing with a large number of trials.

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Most popular questions from this chapter

Suppose you are conducting a binomial experiment that has 15 trials and the probability of success of .02. According to the sample size requirements, you cannot use the normal distribution to approximate the binomial distribution in this situation. Use the mean and standard deviation of this binomial distribution and the empirical rule to explain why there is a problem in this situation. (Note: Drawing the graph and marking the values that correspond to the empirical rule is a good way to start.)

A variation of a roulette wheel has slots that are not of equal size. Instead, the width of any slot is proportional to the probability that a standard normal random variable \(z\) takes on a value between \(a\) and \((a+.1)\), where \(a=-3.0,-2.9,-2.8, \ldots, 2.9,3.0 .\) In other words, there are slots for the intervals \((-3.0,-2.9),(-2.9,-2.8),(-2.8,-2.7)\) through \((2.9,3.0)\). There is one more slot that represents the probability that \(z\) falls outside the interval \((-3.0,3.0)\). Find the following probabilities. a. The ball lands in the slot representing \((.3, .4)\). b. The ball lands in any of the slots representing \((-.1, .4)\). c. In at least one out of five games, the ball lands in the slot representing \((-.1, .4)\). d. In at least 100 out of 500 games, the ball lands in the slot representing \((.4, .5)\).

Let \(x\) be a continuous random variable. What is the probability that \(x\) assumes a single value, such as \(a\) ?

Find the area under the standard normal curve a. to the right of \(z=1.36\) b. to the left of \(z=-1.97\) \(c .\) to the right of \(z=-2.05\) d. to the left of \(z=1.76\)

Obtain the area under the standard normal curve a. to the right of \(z=1.43\) b. to the left of \(z=-1.65\) c. to the right of \(z=-.65\) d. to the left of \(z=.89\)
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