8.1 A Confidence Interval When the Population Standard Deviation Is Known or Large Sample Size
3 min read•june 25, 2024
help estimate population means using sample data. They're crucial when we know the population's or have a large . allows us to create these intervals, giving us a range of likely values for the true .
Sample size and impact interval width. Larger samples give more precise estimates, while higher confidence levels widen intervals. Researchers must balance precision and certainty when choosing confidence levels, considering the tradeoffs between narrow and wide intervals.
Confidence Intervals for Population Mean with Known Standard Deviation or Large Sample Size
Confidence intervals using Central Limit Theorem
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- enables creating for population mean () when () is known or sample size is large ()
- Confidence interval formula:
- represents ()
- represents from based on desired confidence level
- represents known population standard deviation
- represents sample size
- Find critical value () using table or calculator
- : and
- : and
- Confidence interval provides range of plausible values for population mean based on sample data
- Example: Estimating average height of students in a school with known standard deviation of 5 cm and sample mean of 170 cm () at 95% confidence level
Effects of sample size on intervals
- Sample size () impacts width of confidence interval
- Increasing sample size decreases width of confidence interval
- Larger sample sizes yield more precise estimates of population mean
- Example: Doubling sample size from 50 to 100 students in height example
- Interval width reduced from 2.8 cm to 2.0 cm
- quantifies range of values added and subtracted from sample mean to create confidence interval
- Margin of error formula:
- Increasing sample size or decreasing confidence level reduces margin of error, resulting in narrower interval
Tradeoffs in confidence level vs width
- Inverse relationship exists between confidence level and interval width
- Increasing confidence level widens interval, while decreasing confidence level narrows interval
- Higher confidence levels (99%) provide more certainty that true population mean falls within interval
- Increased certainty comes at cost of wider, less precise interval
- Lower confidence levels (90%) result in narrower intervals, providing more precise estimate of population mean
- Increased precision comes with lower confidence that true population mean falls within interval
- Researchers must balance desired confidence level with need for precision when selecting confidence level for study
- Example: Comparing 90%, 95%, and 99% confidence intervals for student height example ()
- 90% CI:
- 95% CI:
- 99% CI:
Statistical Concepts and Confidence Intervals
- forms the basis for constructing confidence intervals
- measures the number of standard deviations a data point is from the mean
- uses confidence intervals to make inferences about population parameters
- affect the shape of the sampling distribution for small sample sizes
Key Terms to Review (35)
$ ext{sigma}$: $ ext{sigma}$ is the Greek letter used to represent the population standard deviation, which is a measure of the spread or dispersion of a population's values around the population mean. It is a crucial statistical concept that is particularly relevant in the context of constructing confidence intervals when the population standard deviation is known or the sample size is large.
$ ext{sqrt}(n)$: $ ext{sqrt}(n)$ is the square root of the sample size, $n$. It is a key statistical concept used in the context of confidence intervals when the population standard deviation is known or the sample size is large.
$\alpha$: $\alpha$ is a statistical concept that represents the probability of making a Type I error, which is the error of rejecting a null hypothesis when it is actually true. It is a crucial parameter in the context of hypothesis testing and confidence interval construction, particularly when the population standard deviation is known or the sample size is large.
$\mu$: $\mu$ (pronounced 'mew') is the symbol used to represent the population mean, which is the average value of a characteristic or variable within a entire population. This term is particularly relevant in the context of 8.1 A Confidence Interval When the Population Standard Deviation Is Known or Large Sample Size, as the population mean is a crucial parameter in constructing confidence intervals.
$ar{x}$: $ar{x}$ is the sample mean, which represents the average value of a random variable within a sample drawn from a population. It is a crucial statistic used in statistical inference, particularly in the context of constructing confidence intervals when the population standard deviation is known or the sample size is large.
$z_{\alpha/2}$: $z_{\alpha/2}$ is the critical value of the standard normal distribution that corresponds to a given significance level, $\alpha$. It is a crucial concept in statistical inference, particularly in the context of constructing confidence intervals and hypothesis testing.
95% Confidence Level: A 95% confidence level is a statistical measure that indicates the range of values within which the true population parameter is likely to fall. It represents the probability that the estimated value from a sample will be within a certain range of the true population value, in this case, 95% of the time.
99% Confidence Level: A 99% confidence level is a statistical measure that indicates there is a 99% probability that a population parameter falls within a given range or interval. It is commonly used in hypothesis testing and interval estimation to make inferences about a population based on a sample.
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.
Confidence intervals: Confidence intervals are ranges of values used to estimate a population parameter with a certain level of confidence. They provide an interval within which the true value of the parameter is expected to fall.
Confidence Intervals: A confidence interval is a range of values that is likely to contain an unknown population parameter, such as the mean or proportion, with a specified level of confidence. It provides a way to quantify the uncertainty around a point estimate and make inferences about the true value of the parameter in the population.
Confidence Level: Confidence level is a statistical measure that quantifies the degree of certainty associated with a sample estimate or a hypothesis test. It represents the probability that the true parameter value falls within a specified range or interval, given the observed data.
Critical Value: The critical value is a threshold value used in hypothesis testing and confidence interval construction to determine whether the observed data is statistically significant or not. It represents the boundary between the region where the null hypothesis is accepted and the region where it is rejected, based on the chosen level of significance.
Critical values: Critical values are specific points in a probability distribution that mark the boundaries for rejecting or failing to reject a null hypothesis. They correspond to the chosen significance level $(\alpha)$ and are used to determine whether a test statistic falls within the rejection region.
Degrees of freedom: Degrees of freedom refer to the number of independent values or quantities which can be assigned to a statistical distribution. They are crucial in estimating population parameters and conducting hypothesis tests.
Degrees of Freedom: Degrees of freedom (df) is a statistical concept that represents the number of values in a data set that are free to vary after certain restrictions or constraints have been imposed. It is a crucial parameter in various statistical analyses and tests, as it determines the appropriate probability distributions and the precision of estimates.
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.
Hypothesis Testing: Hypothesis testing is a statistical method used to determine whether a claim or hypothesis about a population parameter is supported by the 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 based on the evidence provided by the sample.
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 sample proportion. It represents the range of values around the sample statistic within which the true population parameter is expected to fall with a given level of confidence.
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.
Point Estimate: A point estimate is a single numerical value used to represent an unknown population parameter, such as the population mean or proportion. It serves as the best guess or the most likely value for the true parameter based on the available sample data.
Population Mean: The population mean, denoted by the Greek letter μ, is the average or central value of a population. It represents the typical or central tendency of the entire population being studied, rather than just a sample drawn from that population. The population mean is a crucial concept in statistics, probability, and various statistical analyses, as it provides important insights into the characteristics and behavior of a population.
Population Standard Deviation: The population standard deviation is a measure of the spread or dispersion of a set of data within a population. It represents the average distance of each data point from the population mean, and is a fundamental concept in statistics that is closely related to the topics of Definitions of Statistics, Probability, and Key Terms, Measures of the Spread of the Data, A Confidence Interval When the Population Standard Deviation Is Known or Large Sample Size, Calculating the Sample Size n: Continuous and Binary Random Variables, and Probability Distribution Needed for Hypothesis Testing.
Sample Mean: The sample mean, also known as the arithmetic mean, is a measure of central tendency that represents the average value of a set of observations or data points drawn from a population. It is a fundamental concept in statistics that is widely used in various statistical analyses and inferences.
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.
Sampling Distribution: The sampling distribution is a probability distribution that describes the possible values of a statistic, such as the sample mean or sample proportion, obtained from all possible samples of the same size drawn from a population. It represents the distribution of a statistic across all possible samples, rather than the distribution of the population itself.
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 error: Standard error measures the accuracy with which a sample distribution represents a population by using standard deviation. It is crucial for estimating population parameters and conducting hypothesis tests.
Standard Error: The standard error is a measure of the variability or spread of a sample statistic, such as the sample mean. It represents the standard deviation of the sampling distribution of a statistic, indicating how much the statistic is expected to vary from one sample to another drawn from the same population.
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.
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.
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.