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6.2: The Standard Normal Distribution
https://stats.libretexts.org/Courses/Fresno_City_College/Introduction_to_Business_Statistics_-_OER_-_Spring_2023/06%3A_The_Normal_Distribution/6.02%3A_The_Standard_Normal_Distribution
The z-score tells you how many standard deviations the value $\bf{x}$ is above (to the right of) or below (to the left of) the mean, $\bf{\mu}$ .Values of $x$ that are larger than the mean have p...The z-score tells you how many standard deviations the value $\bf{x}$ is above (to the right of) or below (to the left of) the mean, $\bf{\mu}$ .Values of $x$ that are larger than the mean have positive z-scores, and values of $x$ that are smaller than the mean have negative z-scores. About 95% of the $x$ values lie between $–2\sigma$ and $+2\sigma$ of the mean $\mu$ (within two standard deviations of the mean).
4.8.1: The Standard Normal Distribution
https://stats.libretexts.org/Courses/Saint_Mary's_College_Notre_Dame/BFE_1201_Statistical_Methods_for_Finance_(Kuter)/04%3A_Random_Variables/4.08%3A_Introduction_to_Normal_Distribution/4.8.01%3A_The_Standard_Normal_Distribution
The z-score tells you how many standard deviations the value $\bf{x}$ is above (to the right of) or below (to the left of) the mean, $\bf{\mu}$ . About 95% of the $x$ values lie between \(–2\sigm...The z-score tells you how many standard deviations the value $\bf{x}$ is above (to the right of) or below (to the left of) the mean, $\bf{\mu}$ . About 95% of the $x$ values lie between $–2\sigma$ and $+2\sigma$ of the mean $\mu$ (within two standard deviations of the mean). About 99.7% of the $x$ values lie between $–3\sigma$ and $+3\sigma$ of the mean $\mu$ (within three standard deviations of the mean).
1.3: Probability
https://stats.libretexts.org/Bookshelves/Introductory_Statistics/Statistics_Through_an_Equity_Lens_(Anthony)/01%3A_Chapters/1.03%3A_Probability
This chapter introduces the concept of probability, emphasizing its significance in daily decision-making and its role in statistics. It covers probability definitions, rules, distributions, and their...This chapter introduces the concept of probability, emphasizing its significance in daily decision-making and its role in statistics. It covers probability definitions, rules, distributions, and their application to social justice issues. Key topics include the normal curve, z-scores, and blending probability with social issues like racial profiling and wrongful convictions. The chapter underscores the importance of understanding probability to facilitate awareness and drive social change.
6.1: The Standard Normal Distribution
https://stats.libretexts.org/Bookshelves/Applied_Statistics/Business_Statistics_(OpenStax)/06%3A_The_Normal_Distribution/6.01%3A_The_Standard_Normal_Distribution
This page explains the standard normal distribution, defined by z-scores that indicate a value's distance from the mean (0) in terms of standard deviations (1). Z-scores can be positive or negative, a...This page explains the standard normal distribution, defined by z-scores that indicate a value's distance from the mean (0) in terms of standard deviations (1). Z-scores can be positive or negative, and the Empirical Rule highlights that approximately 68%, 95%, and 99.7% of values fall within 1, 2, and 3 standard deviations from the mean. Examples provided illustrate these concepts with specific normal distributions.
6.2: The Standard Normal Distribution
https://stats.libretexts.org/Courses/Fresno_City_College/Book%3A_Business_Statistics_Customized_(OpenStax)/06%3A_The_Normal_Distribution/6.02%3A_The_Standard_Normal_Distribution
The z-score tells you how many standard deviations the value $\bf{x}$ is above (to the right of) or below (to the left of) the mean, $\bf{\mu}$ .Values of $x$ that are larger than the mean have p...The z-score tells you how many standard deviations the value $\bf{x}$ is above (to the right of) or below (to the left of) the mean, $\bf{\mu}$ .Values of $x$ that are larger than the mean have positive z-scores, and values of $x$ that are smaller than the mean have negative z-scores. About 95% of the $x$ values lie between $–2\sigma$ and $+2\sigma$ of the mean $\mu$ (within two standard deviations of the mean).
9.4: Applications of Normal Distributions
https://stats.libretexts.org/Courses/Fullerton_College/Math_100%3A_Liberal_Arts_Math_(Ikeda)/09%3A_Normal_Distribution/9.04%3A_Applications_of_Normal_Distributions
The normal distribution is the foundation for statistical inference and will be an essential part of many of those topics in later chapters. In the meantime, this section will cover some of the types ...The normal distribution is the foundation for statistical inference and will be an essential part of many of those topics in later chapters. In the meantime, this section will cover some of the types of questions that can be answered using the properties of a normal distribution. The first examples deal with more theoretical questions that will help you master basic understandings and computational skills, while the later problems will provide examples with real data, or at least a real context.
4.12: z-scores
https://stats.libretexts.org/Bookshelves/Applied_Statistics/Answering_Questions_with_Data_-__Introductory_Statistics_for_Psychology_Students_(Crump)/04%3A_Probability_Sampling_and_Estimation/4.12%3A_z-scores
library(ggplot2) dnorm_vec <- dnorm(seq(-5,5,.1),mean=0,sd=1) x_range <- seq(-5,5,.1) t_df<-data.frame(x_range,dnorm_vec) ggplot(t_df, aes(x=x_range,y=dnorm_vec))+ geom_line()+ geom_vline(xintercept =...library(ggplot2) dnorm_vec <- dnorm(seq(-5,5,.1),mean=0,sd=1) x_range <- seq(-5,5,.1) t_df<-data.frame(x_range,dnorm_vec) ggplot(t_df, aes(x=x_range,y=dnorm_vec))+ geom_line()+ geom_vline(xintercept = 0)+ geom_vline(xintercept = c(-3,-2,-1,1,2,3))+ theme_classic()+ ylab("Density")+ xlab("score") + scale_x_continuous(breaks=seq(-5,5,1))+ geom_label(data = data.frame(x=-.5, y=.3, label=round(pnorm(c(0,1),0,1)[2]-pnorm(c(0,1),0,1)[1], digits=3)), aes(x = x, y = y, label = label))+ geom_label(data …

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