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Chapter 7: Problem 64

Access the applet Normal Approximation to Binomial Distribution (at https://college.cengage.com/nextbook/statistics/wackerly 966371/student/html/index.html). When the applet is started, it displays the details in Example 7.11 and Figure 7.9 . Initially, the display contains only the binomial histogram and the exact value (calculated using the binomial probability function) for \(p(8)=P(Y=8) .\) Scroll down a little and click the button "Toggle Normal Approximation" to overlay the normal density with mean 10 and standard deviation \(\sqrt{6}=2.449\) the same mean and standard deviation as the binomial random variable \(Y\). You will get a graph superior to the one in Figure \(7.9 .\) a. How many probability mass or density functions are displayed? b. Enter 0 in the box labeled "Begin" and press the enter key. What probabilities do you obtain? c. Refer to part (b). On the line where the approximating normal probability is displayed, you see the expression Normal: $$ P(-0.5<=k<=8.5)=0.2701 $$ Why are the .5s in this expression?

Short Answer

Expert verified
a. Two functions are displayed. b. Probabilities are given for \(k=0\). c. The \.5\s adjust for continuity in a discrete-to-continuous approximation.

Step by step solution

01

Launch the Applet

Navigate to the provided URL and start the applet. Initially, you'll see the binomial histogram for Example 7.11, focusing on the event with the probability \( P(Y=8) \).
02

Overlay the Normal Approximation

Use the "Toggle Normal Approximation" button to overlay the normal distribution curve. This will allow you to compare the binomial distribution with its normal approximation.
03

Analyze the Probabilities

After toggling the normal approximation, analyze the displayed distributions. You should see two: the original binomial distribution and the overlayed normal density function.
04

Enter Begin Value and Observe Probabilities

Enter '0' in the box labeled "Begin" and press Enter. This will update the applet to provide probabilities under the curve for the range starting from 0.
05

Understanding the Normal Expression

The expression \( P(-0.5 \leq k \leq 8.5) = 0.2701 \) includes \(0.5\) due to the continuity correction. This correction is used when approximating a discrete distribution, like the binomial, with a continuous distribution, like the normal, to improve accuracy.

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Key Concepts

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

Probability Mass Function
The probability mass function (PMF) is a critical concept in understanding the binomial distribution. It provides us with the probability that a discrete random variable is exactly equal to some value. Since the binomial distribution focuses on discrete outcomes, the PMF helps calculate the likelihood of each specific result occurring.
For instance, if we are flipping a coin, and want to find the probability of getting heads on exactly 3 out of 6 flips, the PMF will provide this probability, given set parameters such as the number of trials and probability of success for each trial.
The PMF of a binomial distribution can be expressed with the formula: \[ P(Y=k) = \binom{n}{k} p^k (1-p)^{n-k} \] where
  • \( n \) = number of trials
  • \( k \) = number of successful trials
  • \( p \) = probability of success in a single trial
Understanding this formula is crucial for calculating exact probabilities of binomial outcomes.
Continuity Correction
When approximating a discrete distribution using a continuous distribution, such as in the Normal Approximation to Binomial Distribution, continuity correction plays an important role. The binomial distribution is inherently discrete, showing distinct and separate outcomes. However, the normal distribution is continuous, representing a smooth curve.
To bridge this gap and make the approximation more accurate, a continuity correction is applied. This involves adjusting the discrete variable by \(0.5\) to better align with the continuous model.
For example, to determine the probability of exactly 8 successes in a binomial context using normal approximation, continuity correction would adjust the endpoints: instead of using \( P(k=8) \), it becomes \( P(7.5 \leq k \leq 8.5) \). It's this adjustment that explains the expression seen in the problem solution \( P(-0.5 \leq k \leq 8.5) \).
Continuity correction is vital whenever one uses a normal distribution to approximate a binomial distribution, ensuring the results are as accurate as possible.
Binomial Distribution
Binomial Distribution is a fundamental concept in statistics used to model the number of successes in a fixed number of experiments or trials. Each trial is independent and has only two outcomes: success or failure.
The binomial distribution is defined by two parameters:
  • \( n \): the number of trials
  • \( p \): the probability of success on an individual trial
For example, if you are conducting 10 trials with a success probability of 0.5 on each trial, this scenario is modeled using a binomial distribution.
This distribution helps in understanding the probability of a given number of successes over a series of trials, which can be crucial in various fields such as quality control, finance, and anywhere that requires decision-making based on uncertain or probabilistic events.
Mathematically, the mean and standard deviation for a binomial distribution are: \( \mu = np \) and \( \sigma = \sqrt{np(1-p)} \). These parameters help in approximating the binomial distribution to a normal distribution when the conditions are appropriate, usually when \( n \) is large and \( p \) is not too close to 0 or 1.

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Most popular questions from this chapter

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Resistors to be used in a circuit have average resistance 200 ohms and standard deviation 10 ohms. Suppose 25 of these resistors are randomly selected to be used in a circuit. a. What is the probability that the average resistance for the 25 resistors is between 199 and 202 ohms? b. Find the probability that the total resistance does not exceed 5100 ohms. [Hint: see Example 7.9.]
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