Confidence interval for a mean
In interpretCI: Estimate the Confidence Interval and Interpret Step by Step

knitr::opts_chunk$set(echo = TRUE,comment=NA)
library(interpretCI)
library(glue)

#x<-params$result
#x=meanCI(mtcars$mpg)

xresult=x$result[1,]

two.sided<-greater<-less<-FALSE
if(xresult$alternative=="two.sided") two.sided=TRUE
if(xresult$alternative=="less") less=TRUE
if(xresult$alternative=="greater") greater=TRUE

twoS="The null hypothesis will be rejected if the sample mean is too big or if it is too small."
lessS="The null hypothesis will be rejected if the sample mean is too small."
greaterS="The null hypothesis will be rejected if the sample mean is too big."

This document is prepared automatically using the following R command.

call=paste0(deparse(x$call),collapse="")
x1=paste0("library(interpretCI)\nx=",call,"\ninterpret(x)")
textBox(x1,italic=TRUE,bg="grey95",lcolor="grey50")

Problem

string=glue("An inventor has developed a new, energy-efficient lawn mower engine. From his stock of  {xresult$n*100} engines, the inventor selects a simple random sample of {xresult$n} engines for testing. The engines run for an average of {round(xresult$m,2)} minutes on a single gallon of regular gasoline, with a standard deviation of {round(xresult$s,2)} minutes. What is the {(1-xresult$alpha)*100}% confidence interval for the average minutes? (Assume that run times for the population of engines are normally distributed.")

textBox(string)

Confidence interval of mean

The approach that we used to solve this problem is valid when the following conditions are met.

The sampling method must be simple random sampling.
The sampling distribution should be approximately normally distributed.

Since the above requirements are satisfied, we can use the following four-step approach to construct a confidence interval of mean.

Raw data

r ifelse(is.na(x$data[1,1]),"Raw data is not provided.","The first 10 rows of the provided data is as follows.")

if(!is.na(x$data[1,1])) {
     head(x$data,10)
}

Sample statistics

The sample size is r xresult$n, the sample mean is r round(xresult$m,2) and the standard error of sample is r round(xresult$s,2). The confidence level is r (1-xresult$alpha)*100 %.

Find the margin of error

Since we do not know the standard deviation of the population, we cannot compute the standard deviation of the sample mean; instead, we compute the standard error (SE). Because the sample size is much smaller than the population size, we can use the "approximate" formula for the standard error.

$$ SE= \frac{s}{\sqrt{n}}$$ where s is the standard deviation of the sample, n is the sample size.

$$SE=\frac{r round(xresult$s,2)}{\sqrt{r xresult$n}}=r round(xresult$se,2)$$ Find the critical probability(p*):

if(xresult$alternative=="two.sided"){
    string=glue("$$p*=1-\\alpha/2=1-{xresult$alpha}/2={1- xresult$alpha/2}$$")
} else{
     string=glue("$$p*=1-\\alpha=1-{xresult$alpha}$$")
}

r string

The degree of freedom(df) is: $$df=n-1=r xresult$n-1=r xresult$DF$$

The critical value is the t statistic having r xresult$DF degrees of freedom and a cumulative probability equal to r ifelse(xresult$alternative=="two.sided",1- xresult$alpha/2,1- xresult$alpha). From the t Distribution table, we find that the critical value is r round(xresult$critical,3).

show_t_table(DF=xresult$DF,p=xresult$alpha,alternative=xresult$alternative)

if(xresult$alternative=="two.sided"){
  string=glue("$$qt(p,df)=qt({1- xresult$alpha/2},{xresult$DF})={round(xresult$critical,3)}$$")
} else {
    string=glue("$$qt(p,df)=qt({1- xresult$alpha},{xresult$DF})={round(xresult$critical,3)}$$") 
}

r string

The graph shows the $\alpha$ values are the tail areas of the distribution.

draw_t(DF=xresult$DF,p=xresult$alpha,alternative=xresult$alternative)

Compute margin of error(ME):

$$ME=critical\ value \times SE$$ $$ME=r round(xresult$critical,3) \times r round(xresult$se,3)=r round(xresult$ME,3)$$

if(two.sided) {
     string="The range of the confidence interval is defined by the sample statistic $\\pm$margin of error."
} else if(less){
     string="The range of the confidence interval is defined by the -$\\infty$(infinite) and the sample statistic + margin of error."
} else{
     string="The range of the confidence interval is defined by the sample statistic - margin of error and the $\\infty$(infinite)."
}

Specify the confidence interval. r string And the uncertainty is denoted by the confidence level.

Confidence interval of the mean

```{glue,results='asis',echo=FALSE} Therefore, the {(1-xresult$alpha)100}% confidence interval is {round(xresult$lower,2)} to {round(xresult$upper,2)}. That is, we are {(1-xresult$alpha)100}% confident that the true population mean is in the range {round(xresult$lower,2)} to {round(xresult$upper,2)}.

### Plot

You can visualize the mean difference:

```r
plot(x)

Result of meanCI()

print(x)

Reference

The contents of this document are modified from StatTrek.com. Berman H.B., "AP Statistics Tutorial", [online] Available at: https://stattrek.com/estimation/confidence-interval-mean.aspx?tutorial=AP URL[Accessed Data: 1/23/2022].