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 --- title: "Exploring Distributions" output: rmarkdown::html_vignette vignette: > %\VignetteIndexEntry{Exploring Distributions} %\VignetteEngine{knitr::rmarkdown} %\VignetteEncoding{UTF-8} --- ```{r, echo=FALSE, message=FALSE} library(vistributions) ``` In exploring statistical distributions, we focus on the following: - what influences the shape of a distribution - calculate probability from a given quantile - calculate quantiles out of given probability To explore the above 3 concepts, we have defined functions for the following distributions: - Normal - Binomial - Chi Square - F - t ## Normal Distribution ### Distribution Shape Visualize how changes in mean and standard deviation affect the shape of the normal distribution. ##### Input - mean: mean of the normal distribution - sd: standard deviation of the normal distribution ##### Output - Normal distribution plot ```{r norm_plot, fig.width=7, fig.height=7, fig.align='centre'} vdist_normal_plot(mean = 2, sd = 0.6) ``` ### Percentiles #### Calculate and visualize quantiles out of given probability. ##### Input - probs: a probability value - mean: mean of the normal distribution - sd: standard deviation of the normal distribution - type: lower/upper tail Suppose X, the grade on a exam, is normally distributed with mean 60 and standard deviation 3. The teacher wants to give 10% of the class an A. What should be the cutoff to determine who gets an A? ```{r norm_per1, fig.width=7, fig.height=7, fig.align='centre'} vdist_normal_perc(0.10, 60, 3, 'upper') ``` The teacher wants to give lower 15% of the class a D. What cutoff should the teacher use to determine who gets an D? ```{r norm_per2, fig.width=7, fig.height=7, fig.align='centre'} vdist_normal_perc(0.85, 60, 3, 'lower') ``` The teacher wants to give middle 50% of the class a B. What cutoff should the teacher use to determine who gets an B? ```{r norm_per3, fig.width=7, fig.height=7, fig.align='centre'} vdist_normal_perc(0.5, 60, 3, 'both') ``` ### Probabilities #### Calculate and visualize probability from a given quantile ##### Input - perc: a quantile value - mean: mean of the normal distribution - sd: standard deviation of the normal distribution - type: lower/upper/both tail Let X be the IQ of a randomly selected student of a school. Assume X ~ N(90, 4). What is the probability that a randomly selected student has an IQ below 80? ```{r norm_prob1, fig.width=7, fig.height=7, fig.align='centre'} vdist_normal_prob(80, mean = 90, sd = 4) ``` What is the probability that a randomly selected student has an IQ above 100? ```{r norm_prob2, fig.width=7, fig.height=7, fig.align='centre'} vdist_normal_prob(100, mean = 90, sd = 4, type = 'upper') ``` What is the probability that a randomly selected student has an IQ between 85 and 100? ```{r norm_prob3, fig.width=7, fig.height=7, fig.align='centre'} vdist_normal_prob(c(85, 100), mean = 90, sd = 4, type = 'both') ``` ## Binomial Distribution ### Distribution Shape Visualize how changes in number of trials and the probability of success affect the shape of the binomial distribution. ```{r binom_plot, fig.width=7, fig.height=7, fig.align='centre'} vdist_binom_plot(10, 0.3) ``` ### Percentiles #### Calculate and visualize quantiles out of given probability ##### Input - p: a single aggregated probability of multiple trials - n: the number of trials - tp: the probability of success in a trial - type: lower/upper tail ```{r binom_per1, fig.width=7, fig.height=7, fig.align='centre'} vdist_binom_perc(10, 0.5, 0.05) ``` ```{r binom_per2, fig.width=7, fig.height=7, fig.align='centre'} vdist_binom_perc(10, 0.5, 0.05, 'upper') ``` ### Probabilities #### Calculate and visualize probability from a given quantile ##### Input - p: probability of success - n: the number of trials - s: number of success in a trial - type: lower/upper/interval/exact tail Assume twenty-percent (20%) of Magemill have no health insurance. Randomly sample n = 12 Magemillians. Let X denote the number in the sample with no health insurance. What is the probability that exactly 4 of the 15 sampled have no health insurance? ```{r binom_prob1, fig.width=7, fig.height=7, fig.align='centre'} vdist_binom_prob(12, 0.2, 4, type = 'exact') ``` What is the probability that at most one of those sampled has no health insurance? ```{r binom_prob2, fig.width=7, fig.height=7, fig.align='centre'} vdist_binom_prob(12, 0.2, 1, 'lower') ``` What is the probability that more than seven have no health insurance? ```{r binom_prob3, fig.width=7, fig.height=7, fig.align='centre'} vdist_binom_prob(12, 0.2, 8, 'upper') ``` What is the probability that fewer than 5 have no health insurance? ```{r binom_prob4, fig.width=7, fig.height=7, fig.align='centre'} vdist_binom_prob(12, 0.2, c(0, 4), 'interval') ``` ## Chi Square Distribution ### Distribution Shape Visualize how changes in degrees of freedom affect the shape of the chi square distribution. ```{r chi_plot, fig.width=7, fig.height=7, fig.align='centre'} vdist_chisquare_plot(df = 5) vdist_chisquare_plot(df = 5, normal = TRUE) ``` ### Percentiles #### Calculate quantiles out of given probability ##### Input - probs: a probability value - df: degrees of freedom - type: lower/upper tail Let X be a chi-square random variable with 8 degrees of freedom. What is the upper fifth percentile? ```{r chi_per1, fig.width=7, fig.height=7, fig.align='centre'} vdist_chisquare_perc(0.05, 8, 'upper') ``` What is the tenth percentile? ```{r chi_per2, fig.width=7, fig.height=7, fig.align='centre'} vdist_chisquare_perc(0.10, 8, 'lower') ``` ### Probability #### Calculate probability from a given quantile. ##### Input - perc: a quantile value - df: degrees of freedom - type: lower/upper tail What is the probability that a chi-square random variable with 12 degrees of freedom is greater than 8.79? ```{r chi_prob1, fig.width=7, fig.height=7, fig.align='centre'} vdist_chisquare_prob(8.79, 12, 'upper') ``` What is the probability that a chi-square random variable with 12 degrees of freedom is greater than 8.62? ```{r chi_prob2, fig.width=7, fig.height=7, fig.align='centre'} vdist_chisquare_prob(8.62, 12, 'lower') ``` ## F Distribution ### Distribution Shape Visualize how changes in degrees of freedom affect the shape of the F distribution. ```{r f_plot, fig.width=7, fig.height=7, fig.align='centre'} vdist_f_plot() vdist_f_plot(6, 10, normal = TRUE) ``` ### Percentiles #### Calculate quantiles out of given probability ##### Input - probs: a probability value - num_df: nmerator degrees of freedom - den_df: denominator degrees of freedom - type: lower/upper tail Let X be an F random variable with 4 numerator degrees of freedom and 5 denominator degrees of freedom. What is the upper twenth percentile? ```{r f_per1, fig.width=7, fig.height=7, fig.align='centre'} vdist_f_perc(0.20, 4, 5, 'upper') ``` What is the 35th percentile? ```{r f_per2, fig.width=7, fig.height=7, fig.align='centre'} vdist_f_perc(0.35, 4, 5, 'lower') ``` ### Probabilities #### Calculate probability from a given quantile. ##### Input - perc: a quantile value - num_df: nmerator degrees of freedom - den_df: denominator degrees of freedom - type: lower/upper tail What is the probability that an F random variable with 4 numerator degrees of freedom and 5 denominator degrees of freedom is greater than 3.89? ```{r f_prob1, fig.width=7, fig.height=7, fig.align='centre'} vdist_f_prob(3.89, 4, 5, 'upper') ``` What is the probability that an F random variable with 4 numerator degrees of freedom and 5 denominator degrees of freedom is less than 2.63? ```{r f_prob2, fig.width=7, fig.height=7, fig.align='centre'} vdist_f_prob(2.63, 4, 5, 'lower') ``` ## t Distribution ### Distribution Shape Visualize how degrees of freedom affect the shape of t distribution. ```{r t_plot, fig.width=7, fig.height=7, fig.align='centre'} vdist_t_plot(df = 8) ``` ### Percentiles #### Calculate quantiles out of given probability ##### Input - probs: a probability value - df: degrees of freedom - type: lower/upper/both tail What is the upper fifteenth percentile? ```{r t_per1, fig.width=7, fig.height=7, fig.align='centre'} vdist_t_perc(0.15, 8, 'upper') ``` What is the eleventh percentile? ```{r t_per2, fig.width=7, fig.height=7, fig.align='centre'} vdist_t_perc(0.11, 8, 'lower') ``` What is the area of the curve that has 95% of the t values? ```{r t_per3, fig.width=7, fig.height=7, fig.align='centre'} vdist_t_perc(0.8, 8, 'both') ``` ### Probabilities #### Calculate probability from a given quantile. ##### Input - perc: a quantile value - df: degrees of freedom - type: lower/upper/interval/both tail Let T follow a t-distribution with r = 6 df. What is the probability that the value of T is less than 2? ```{r t_prob1, fig.width=7, fig.height=7, fig.align='centre'} vdist_t_prob(2, 6, 'lower') ``` What is the probability that the value of T is greater than 2? ```{r t_prob2, fig.width=7, fig.height=7, fig.align='centre'} vdist_t_prob(2, 6, 'upper') ``` What is the probability that the value of T is between -2 and 2? ```{r t_prob3, fig.width=7, fig.height=7, fig.align='centre'} vdist_t_prob(2, 6, 'both') ``` What is the probability that the absolute value of T is greater than 2? ```{r t_prob4, fig.width=7, fig.height=7, fig.align='centre'} vdist_t_prob(2, 6, 'interval') ```
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