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Chapter 6: Problem 48

Find the following binomial probabilities using the normal approximation. a. \(n=140, \quad p=.45, \quad P(x=67)\) b. \(n=100, \quad p=.55, \quad P(52 \leq x \leq 60)\) c. \(n=90, \quad p=.42, \quad P(x \geq 40)\) d. \(n=104, \quad p=.75, \quad P(x \leq 72)\)

Short Answer

Expert verified
The results are a) 0.7379, b) 0.6249, c) 0.3877, and d) 0.0975.

Step by step solution

01

a. Find \(P(x=67)\)

First, let's find the mean and the standard deviation. Mean, \(\mu=np=140*0.45=63\). Standard deviation, \(\sigma=\sqrt{np(1-p)}=\sqrt{140*0.45*(1-0.45)}=7.07106781187\). Now, construct z-score, \(z=\frac{x+0.5-\mu}{\sigma}=\frac{67.5-63}{7.07106781187}=0.635998728013\). Now, lookup the value of 0.6359 in the z-table, which is 0.7379.
02

b. Find \(P(52 \leq x \leq 60)\)

Mean, \(\mu=np=100*0.55=55\). Standard deviation, \(\sigma=\sqrt{np(1-p)}=\sqrt{100*0.55*(1-0.55)}=4.97288224772\). For \(P(x \leq 60)\), the z-score would be \(z=\frac{x+0.5-\mu}{\sigma}=\frac{60.5-55}{4.97288224772}=1.10554159679\). So \(P(x \leq 60) = 0.8659\). Now, \(P(x \geq 52)\) would have the z-score \(z=\frac{x-0.5-\mu}{\sigma}=\frac{51.5-55}{4.97288224772}=-0.703559402767\). So \(P(x \geq 52) = 0.2410\). We subtract the two probabilities to find \(P(52 \leq x \leq 60)\), so \(P(52 \leq x \leq 60) = 0.8659-0.2410=0.6249\)
03

c. Find \(P(x \geq 40)\)

Mean, \(\mu=np=90*0.42=37.8\). Standard deviation, \(\sigma=\sqrt{np(1-p)}=\sqrt{90*0.42*(1-0.42)}=5.9160797831\). For \(P(x \geq 40)\), the z-score would be \(z=\frac{x-0.5-\mu}{\sigma}=\frac{39.5-37.8}{5.9160797831}=0.287347885566\). So \(P(x \geq 40) = 1 - 0.6123 = 0.3877\)
04

d. Find \(P(x \leq 72)\)

Mean, \(\mu=np=104*0.75=78\). Standard deviation, \(\sigma=\sqrt{np(1-p)}=\sqrt{104*0.75*(1-0.75)}=4.24264068712\). For \(P(x \leq 72)\), the z-score would be \(z=\frac{x+0.5-\mu}{\sigma}=\frac{72.5-78}{4.24264068712}=-1.29553203629\). So \(P(x \leq 72) = 0.0975\)

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Key Concepts

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

Understanding Binomial Distribution
In statistics, the binomial distribution is a probability distribution that summarizes the likelihood of a value to succeed or fail in an experiment or survey that is repeated multiple times.

The distribution is defined by two parameters: \( n \) and \( p \).
  • \( n \) is the number of trials or experiments.
  • \( p \) is the probability of success on an individual trial.
The binomial distribution is useful because it models real-world situations and products where there are two potential outcomes: success or failure. Each trial is independent, and the probability of success stays the same from trial to trial. For example, the binomial distribution can be used to predict the number of heads in a series of 10 coin flips.
Exploring Probability Distributions
Probability distributions are functions that describe how probabilities are distributed over the values of a random variable.

In simple terms, they tell us the probability of each possible outcome in an experiment.
  • A discrete probability distribution is one where the random variable involves discrete data, and values occur at distinct points.
  • A continuous probability distribution is one where the data can take any value within a range.
The binomial distribution discussed earlier is a discrete probability distribution. In contrast, the normal distribution, often used as an approximation for the binomial distribution when \( n \) is large, is continuous.

Understanding these distributions helps us to estimate probabilities quickly and make informed decisions based on data.
What is a Z-Score?
The z-score, or standard score, is a measure that describes a value's relationship to the mean of a group of values.

It is expressed in terms of standard deviations from the mean:
  • A z-score of 0 indicates that the data point's score is identical to the mean score.
  • A z-score of 1.0 indicates a value that is one standard deviation from the mean.
The z-score formula is: \[ z = \frac{x - \mu}{\sigma} \]where:
  • \( x \) is the value of the element
  • \( \mu \) is the mean of the data set
  • \( \sigma \) is the standard deviation of the data set
Z-scores are particularly useful in the normal approximation to the binomial distribution, as they allow us to use the standard normal distribution tables to estimate probabilities.
Calculating Standard Deviation
Standard deviation is a measure of the amount of variation or dispersion in a set of values.

It is a widely used measure in statistics to quantify the extent of variability in a given dataset, distinguishing between data spread or concentration.

The formula for calculating the standard deviation in a binomial distribution is: \[ \sigma = \sqrt{np(1-p)} \]where:
  • \( n \) is the number of trials
  • \( p \) is the probability of success
A low standard deviation implies that the data points tend to be close to the mean, whereas a high standard deviation indicates that the data points are spread over a larger range of values. Understanding standard deviation is critical when interpreting data distributions and making predictions.

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Most popular questions from this chapter

For the standard normal distribution, what is the area within three standard deviations of the mean?

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