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6.3 Estimating the Binomial with the Normal Distribution

2 min read•june 25, 2024

The can be used to approximate when certain conditions are met. This powerful tool simplifies calculations for large sample sizes and moderate probabilities of success, making it invaluable in various fields.

By understanding the requirements and applications of this approximation, you can efficiently estimate probabilities in real-world scenarios. This method bridges the gap between discrete and continuous distributions, offering a practical approach to complex problems.

Estimating the Binomial with the Normal Distribution

Normal approximation of binomial probabilities

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  • Binomial probabilities approximated using when certain conditions met
  • Calculate probabilities using :
    1. Find of binomial distribution μ=np\mu = np ([n](https://www.fiveableKeyTerm:n)[n](https://www.fiveableKeyTerm:n) = , [p](https://www.fiveableKeyTerm:p)[p](https://www.fiveableKeyTerm:p) = probability of success)
    2. Find of binomial distribution σ=np(1−p)\sigma = \sqrt{np(1-p)}
    3. Standardize binomial XX using ###-score_0### Z=X−μσZ = \frac{X - \mu}{\sigma}
    4. Use table or calculator to find probability corresponding to
  • Normal approximation useful for large sample sizes (typically n≥30n \geq 30) and probabilities not too close to 0 or 1 (0.1≤p≤0.90.1 \leq p \leq 0.9)
  • Approximation accurate when np≥5np \geq 5 and n(1−p)≥5n(1-p) \geq 5 (ensures sufficient expected successes and failures)
  • supports the use of normal approximation for large sample sizes

Applications of normal distribution estimates

  • Normal distribution accurately estimates binomial probabilities in various situations
  • Suitable when sample size (nn) large (typically n≥30n \geq 30) and probability of success (pp) not too close to 0 or 1 (0.1≤p≤0.90.1 \leq p \leq 0.9)
  • Product of nn and pp (npnp) and product of nn and (1−p)(1-p) (n(1−p)n(1-p)) must be greater than or equal to 5
  • Conditions satisfied, shape of binomial distribution approximately normal (bell-shaped curve)
  • Applications include quality control (defective products), marketing (customer preferences), and finance (loan defaults)

Symmetry in binomial distributions

  • of binomial distribution depends on probability of success (pp)
    • Perfectly symmetric when p=0.5p = 0.5 (equal chance of success and failure)
    • Becomes more skewed as pp moves away from 0.5 (closer to 0 or 1)
  • Normal approximation most accurate when binomial distribution nearly symmetric
    • More accurate for pp close to 0.5 (balanced outcomes)
    • Less accurate for pp close to 0 or 1, especially for smaller sample sizes (skewed distribution)
  • Highly skewed binomial distributions (pp very close to 0 or 1) may require alternative methods (Poisson approximation) for better accuracy

Properties of the Normal Distribution

  • describes the shape of the normal distribution curve
  • gives the probability of a value falling below a certain point
  • of a normal distribution is equal to its mean
  • measures the spread of the distribution around the mean

Key Terms to Review (37)

Bar graph: A bar graph is a visual representation of data using rectangular bars to compare different categories or groups. The lengths of the bars are proportional to the values they represent.
Binomial Probabilities: Binomial probabilities refer to the likelihood of obtaining a specific number of successes in a fixed number of independent Bernoulli trials, where each trial has only two possible outcomes (success or failure) and the probability of success remains constant across all trials. This concept is particularly relevant when estimating the binomial distribution using the normal distribution, as outlined in the topic 6.3 Estimating the Binomial with the Normal Distribution.
Central Limit Theorem: The central limit theorem is a fundamental concept in probability and statistics that states that the sampling distribution of the mean of a random variable will tend to a normal distribution as the sample size increases, regardless of the underlying distribution of the variable.
Continuity Correction: Continuity correction is a statistical adjustment made when using the normal distribution to approximate a discrete probability distribution, such as the binomial distribution. This correction helps to account for the fact that the normal distribution is continuous while the binomial distribution is discrete, ensuring a more accurate approximation.
Cumulative Distribution Function: The cumulative distribution function (CDF) is a fundamental concept in probability and statistics that describes the probability of a random variable taking a value less than or equal to a given value. It provides a comprehensive way to represent the distribution of a random variable and is closely related to other important statistical concepts such as probability density functions and probability mass functions.
Cumulative distribution function (CDF): A cumulative distribution function (CDF) represents the probability that a continuous random variable takes on a value less than or equal to a specific value. It is an integral of the probability density function (PDF).
De Moivre-Laplace Theorem: The De Moivre-Laplace theorem is a fundamental result in probability theory that establishes the relationship between the binomial distribution and the normal distribution. It provides a way to approximate the binomial distribution with the normal distribution when certain conditions are met.
Equal standard deviations: Equal standard deviations, also known as homoscedasticity, occur when the variability within each group being compared is similar. This is an important assumption for performing One-Way ANOVA.
Estimate of the error variance: Estimate of the error variance is a measure of the variability in the observed values that cannot be explained by the regression model. It is often denoted as $\hat{\sigma}^2$ and calculated as the sum of squared residuals divided by the degrees of freedom.
Expected mean: The expected mean in the context of linear regression is the average value of the response variable predicted by the regression equation for a given set of predictor variables. It represents the central tendency around which individual observations are expected to vary.
Expected value: Expected value is the weighted average of all possible values that a random variable can take on, with weights being their respective probabilities. It provides a measure of the center of the distribution of the variable.
Expected Value: Expected value is a statistical concept that represents the average or central tendency of a probability distribution. It is the weighted average of all possible outcomes, where the weights are the probabilities of each outcome occurring. The expected value provides a measure of the central tendency and is a useful tool for decision-making and analysis in various contexts, including the topics of 3.1 Terminology, 4.1 Hypergeometric Distribution, 4.2 Binomial Distribution, 5.1 Properties of Continuous Probability Density Functions, 5.2 The Uniform Distribution, and 6.3 Estimating the Binomial with the Normal Distribution.
Mean: The mean, also known as the arithmetic mean, is a measure of central tendency that represents the average value in a dataset. It is calculated by summing up all the values in the dataset and dividing by the total number of values. The mean provides a summary statistic that describes the central or typical value in a distribution of data.
N: In statistics, 'n' represents the sample size, which is the number of observations or data points collected from a population for analysis. This key concept is crucial as it impacts the reliability and validity of statistical estimates, influencing the power of hypothesis tests and the precision of confidence intervals.
Normal Approximation: The normal approximation is a statistical technique that allows for the use of the normal distribution to estimate the probability of events in a binomial distribution when certain conditions are met. This concept is particularly relevant in the context of various statistical analyses and hypothesis testing.
Normal distribution: A normal distribution is a continuous probability distribution that is symmetrical and bell-shaped, where most of the observations cluster around the central peak. It is characterized by its mean ($\mu$) and standard deviation ($\sigma$).
Normal Distribution: The normal distribution, also known as the Gaussian distribution, is a continuous probability distribution that is symmetrical and bell-shaped. It is one of the most widely used probability distributions in statistics and plays a crucial role in various statistical analyses and concepts covered in this course.
Normal Probability Formula: The normal probability formula, also known as the Gaussian distribution or bell curve, is a mathematical equation that describes the distribution of a continuous random variable. It is widely used in statistical analysis to model and predict the probability of events occurring within a normal distribution.
P: The probability of a single event occurring in a Bernoulli trial, where the event can have one of two possible outcomes (success or failure). The value of 'p' represents the likelihood of the desired outcome (success) happening in a single trial. It is a fundamental parameter in the Binomial and Geometric probability distributions, as well as in the estimation of sample size for continuous and binary random variables.
Probability: Probability is the measure of the likelihood of an event occurring. It quantifies the chance or odds of a particular outcome happening within a given set of circumstances or a defined sample space. Probability is a fundamental concept in statistics, as it provides the foundation for understanding and analyzing uncertainty, risk, and decision-making.
Probability density function: A probability density function (PDF) describes the likelihood of a continuous random variable taking on a particular value. It is represented by a curve where the area under the curve within a given interval represents the probability that the variable falls within that interval.
Probability Density Function: A probability density function (PDF) is a mathematical function that describes the relative likelihood of a continuous random variable taking on a specific value. It provides a way to represent the distribution of a continuous random variable and is a fundamental concept in probability and statistics.
Random variable: A random variable is a numerical outcome of a random phenomenon. It can take on different values, each with an associated probability.
Random Variable: A random variable is a numerical quantity whose value is determined by the outcome of a random phenomenon. It is a variable that can take on different values with certain probabilities, allowing for the quantification of uncertainty and the analysis of random processes.
Sample Size: Sample size refers to the number of observations or data points collected in a statistical study or experiment. It is a crucial factor that determines the reliability and precision of the conclusions drawn from the data.
Sigma Notation (Σ): Sigma notation, denoted by the Greek letter Σ, is a concise way to represent the sum of a series of values or the application of a mathematical operation across multiple elements. It is a fundamental concept in statistics and various mathematical disciplines, allowing for the efficient expression and calculation of sums, means, and other statistical measures.
Skewness: Skewness is a measure of the asymmetry or lack of symmetry in the distribution of a dataset. It describes the extent to which a probability distribution or a data set deviates from a symmetric, bell-shaped, or normal distribution.
Standard Deviation: Standard deviation is a measure of the spread or dispersion of a set of data around the mean. It quantifies the typical deviation of values from the average, providing insight into the variability within a dataset.
Standard normal distribution: The standard normal distribution is a normal distribution with a mean of 0 and a standard deviation of 1. It is used as a reference to transform any normal distribution into a standardized form for easier analysis.
Standard Normal Distribution: The standard normal distribution is a probability distribution that describes a normal distribution with a mean of 0 and a standard deviation of 1. It is a fundamental concept in statistics that is used to analyze and make inferences about data that follows a normal distribution.
Symmetry: Symmetry refers to the balanced and proportional arrangement of elements or features around a central axis or point. It is a fundamental concept that is closely tied to the measures of center, skewness, and the normal distribution in statistics.
The Central Limit Theorem: The Central Limit Theorem (CLT) states that the distribution of the sample mean approaches a normal distribution as the sample size grows, regardless of the original population's distribution. This theorem is fundamental in inferential statistics because it allows for making predictions about population parameters.
Variance: Variance is a measure of the spread or dispersion of a dataset, indicating how far each data point deviates from the mean or average value. It is a fundamental statistical concept that quantifies the variability within a distribution and plays a crucial role in various statistical analyses and probability distributions.
Z: Z is a standardized test statistic that follows a standard normal distribution. It is used to estimate the probability of a particular outcome or to determine the significance of a sample statistic in relation to a population parameter.
Z-score: A z-score represents the number of standard deviations a data point is from the mean. It is used to determine how unusual or typical a value is within a normal distribution.
Z-Score: A z-score, also known as a standard score, is a statistical measure that expresses how many standard deviations a data point is from the mean of a dataset. It is a fundamental concept in probability and statistics that is widely used in various statistical analyses and hypothesis testing.
μ (Mu): Mu (μ) is a Greek letter commonly used in statistics to represent the population mean or average. It is a central parameter that describes the central tendency or typical value of a population distribution. Mu is a crucial concept in understanding various statistical measures and distributions covered in this course.
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