Glossary

8.3 A Population Proportion

2 min read•june 25, 2024

Confidence intervals for population proportions help us estimate the true percentage of a characteristic in a population. We use sample data to calculate a range that likely contains the actual proportion, giving us a measure of uncertainty in our estimate.

Understanding these intervals is crucial for making informed decisions based on sample data. We'll learn how to calculate and interpret them, determine appropriate sample sizes, and connect this knowledge to broader concepts in statistical inference and .

Confidence Intervals for Population Proportions

Confidence intervals for proportions

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  • Calculate range of values likely to contain true () at specified confidence level (90%, 95%, 99%)
  • Requires (p^\hat{p}), sample size (nn), and (z∗z^*) based on confidence level
  • Formula: p^±z∗p^(1−p^)n\hat{p} \pm z^* \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} where p^(1−p^)n\sqrt{\frac{\hat{p}(1-\hat{p})}{n}} is of sample proportion
  • Conditions: random representative sample, large enough sample size (n≥30n \geq 30) or population 10 times larger than sample, met (np^≥10n\hat{p} \geq 10 and n(1−p^)≥10n(1-\hat{p}) \geq 10)

Interpretation of proportion intervals

  • Provides plausible range for true proportion based on sample data
  • Example: 95% for voter support (0.52, 0.58) means 95% confident true proportion falls between 0.52 and 0.58
  • Helps make informed decisions and understand uncertainty of estimate
    • Narrow interval indicates more precise estimate
    • Wide interval suggests more uncertainty
    • Overlapping intervals for different groups suggest no significant difference
  • Accounts for in the estimation process

Sample size for proportion estimates

  • Determines sample size needed for desired (maximum expected difference between sample and true proportion)
  • Requires desired confidence level (, z∗z^*), margin of error (EE), and estimate of population proportion (pp, often 0.5 for conservative value)
  • Formula: n=(z∗)2⋅p(1−p)E2n = \frac{(z^*)^2 \cdot p(1-p)}{E^2}, round up to nearest integer
  • can reduce required sample size if population size is known and small relative to sample size

Statistical Inference and Hypothesis Testing

  • Uses sample data () to draw conclusions about a population
  • Hypothesis testing involves comparing observed data to a null hypothesis
  • (α) determines the threshold for rejecting the null hypothesis
  • of a test is the probability of correctly rejecting a false null hypothesis

Key Terms to Review (29)

"OR" Event: An 'OR' event in probability occurs when at least one of multiple events happens. The probability of an 'OR' event is calculated by adding the probabilities of individual events and subtracting the probability of their intersection.
Binomial distribution: A binomial distribution is a discrete probability distribution of the number of successes in a fixed number of independent Bernoulli trials, each with the same probability of success. It is characterized by parameters $n$ (number of trials) and $p$ (probability of success).
Confidence Interval: A confidence interval is a range of values used to estimate the true value of a population parameter, such as a mean or proportion, based on sample data. It provides a measure of uncertainty around the sample estimate, indicating how much confidence we can have that the interval contains the true parameter value.
Descriptive statistics: Descriptive statistics involves summarizing and organizing data to make it easily understandable. It includes measures such as mean, median, mode, range, and standard deviation.
Error bound: Error bound in statistics quantifies the maximum expected difference between a sample estimate and the true population parameter. It provides a range within which the true value is expected to lie, given a certain level of confidence.
Finite Population Correction Factor: The finite population correction factor is a statistical adjustment applied when sampling from a population that is small relative to the overall size of the population. It accounts for the fact that sampling without replacement from a finite population reduces the variability of the sample compared to sampling with replacement from an infinite population.
Hypothesis Testing: Hypothesis testing is a statistical method used to determine whether a claim or hypothesis about a population parameter is likely to be true or false based on sample data. It involves formulating a null hypothesis and an alternative hypothesis, collecting and analyzing sample data, and making a decision to either reject or fail to reject the null hypothesis.
Independence: Independence is a fundamental concept in statistics that describes the relationship between events or variables. When events or variables are independent, the occurrence or value of one does not depend on or influence the occurrence or value of the other. This concept is crucial in understanding probability, statistical inference, and the analysis of relationships between different factors.
Margin of Error: The margin of error is a statistical measure that quantifies the amount of uncertainty or imprecision in a sample statistic, such as the sample mean or proportion. It represents the range of values above and below the sample statistic within which the true population parameter is expected to fall, with a given level of confidence.
Normal Approximation: The normal approximation is a statistical concept that allows for the use of the normal distribution to approximate other probability distributions, particularly the binomial distribution, when certain conditions are met. This approximation is useful in making inferences about population parameters when dealing with large sample sizes or when the underlying distribution is not known.
Normal approximation to the binomial: Normal approximation to the binomial is a method used to approximate the probabilities of a binomial distribution using the normal distribution when the sample size is large and the probability of success is neither very close to 0 nor 1.
Normal distribution: A normal distribution is a continuous probability distribution that is symmetrical around its mean, with a characteristic bell-shaped curve. In a normal distribution, most of the data points are concentrated around the mean, and the probabilities for values further away from the mean taper off equally in both directions.
P-hat: p-hat, also known as the sample proportion, is a point estimate of the population proportion in statistical inference. It represents the proportion or percentage of a characteristic of interest observed in a sample drawn from a population.
Parameter: A parameter is a numerical characteristic of a population, such as a mean or standard deviation. It represents an entire group rather than a sample taken from it.
Parameter: A parameter is a numerical value or characteristic that defines a population or a statistical model. It represents a fixed, unknown quantity that is used to describe the properties of a larger group or system.
Population Proportion: The population proportion is the percentage or fraction of a population that possesses a certain characteristic or attribute. It is a fundamental concept in statistics that is used to make inferences about the larger population based on a sample drawn from that population.
Power: Power is a statistical concept that refers to the ability of a statistical test to detect an effect or difference if it truly exists in the population. It is a measure of the likelihood that a statistical test will reject the null hypothesis when the alternative hypothesis is true.
R: R is a programming language and software environment for statistical computing and graphics. It is widely used in various fields, including statistics, data analysis, and scientific research, due to its powerful capabilities in handling and analyzing data.
Sample Proportion: The sample proportion is a statistic that represents the proportion or percentage of a sample that exhibits a certain characteristic. It is a crucial concept in statistics, as it allows researchers to make inferences about the characteristics of a larger population based on a smaller, representative sample.
Sampling Variability: Sampling variability refers to the natural fluctuations or differences that occur in sample statistics, such as the sample mean or sample proportion, due to the random nature of the sampling process. It reflects the fact that different samples drawn from the same population will likely produce slightly different results, even when the population parameters remain the same.
Significance Level: The significance level, denoted as α (alpha), is the probability of rejecting the null hypothesis when it is true. It represents the maximum acceptable probability of making a Type I error, which is the error of rejecting the null hypothesis when it is actually true. The significance level is a crucial concept in hypothesis testing and statistical inference, as it helps determine the strength of evidence required to draw conclusions about a population parameter or the relationship between variables.
Simple Random Sample: A simple random sample is a type of probability sampling where each individual in the population has an equal chance of being selected for the sample. This sampling method ensures that the selected sample is representative of the overall population, allowing for unbiased statistical inferences to be made.
Standard Error: Standard error is a statistical term that measures the accuracy with which a sample represents a population. It quantifies the variability of sample means from the true population mean, helping to determine how much sampling error exists when making inferences about the population.
StatCrunch: StatCrunch is a web-based statistical software package that allows users to analyze data, create visualizations, and perform a variety of statistical tests. It is particularly useful in the context of topics related to population proportions, comparing population means, and comparing population proportions.
Statistic: A statistic is a numerical value calculated from a sample of data that is used to describe or make inferences about a population. Statisticians use statistics to analyze data, test hypotheses, and draw conclusions in the context of various fields, such as 1.1 Definitions of Statistics, Probability, and Key Terms, 1.2 Data, Sampling, and Variation in Data and Sampling, and 8.3 A Population Proportion.
Stratified Sampling: Stratified sampling is a probability sampling technique where the population is divided into distinct, non-overlapping subgroups or strata based on one or more characteristics, and then a random sample is selected from each stratum. The purpose is to ensure that the sample is representative of the overall population and to potentially increase the precision of estimates.
Success-Failure Condition: The success-failure condition is a fundamental concept in probability and statistics, particularly in the context of Bernoulli trials and the binomial distribution. It describes a scenario where an experiment or observation has only two possible outcomes: success or failure.
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 a particular observation is within a normal distribution.
Z-Score: A z-score is a standardized measure that expresses how many standard deviations a data point is from the mean of a distribution. It allows for the comparison of data points across different distributions by converting them to a common scale.
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