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

knitr::opts_chunk$set(echo = TRUE,comment=NA,fig.width=7,fig.height=5)
library(interpretCI)
library(glue)

x=meanCI(mtcars,mpg,mu=23)

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

Given Problem : `r ifelse(two.sided,"Two","One")`-Tailed Test

string=glue("An inventor has developed a new, energy-efficient lawn mower engine. He claims that the engine will run continuously for {round(x$result$mu,2)} minutes on a single gallon of regular gasoline. From his stock of 2000 engines, the inventor selects a simple random sample of {x$result$n} engines for testing. The engines run for an average of {round(x$result$m,2)} minutes, with a standard deviation of {round(x$result$s,2)} minutes. Test the null hypothesis that the mean run time {ifelse(two.sided,'is',ifelse(less, 'greater than','less than'))} {x$result$mu} minutes against the alternative hypothesis that the mean run time {ifelse(two.sided,'is not',ifelse(less, 'less than','greater than'))} {round(x$result$mu,2)} minutes. Use a {x$result$alpha} level of significance. (Assume that run times for the population of engines are normally distributed.)")

textBox(string)

Hypothesis Test for a Mean

This lesson explains how to conduct a hypothesis test of a mean, when the following conditions are met:

The sampling method is simple random sampling.
The sampling distribution is normal or nearly normal.

Generally, the sampling distribution will be approximately normally distributed if any of the following conditions apply.

The population distribution is normal.
The population distribution is symmetric, unimodal, without outliers, and the sample size is 15 or less.
The population distribution is moderately skewed, unimodal, without outliers, and the sample size is between 16 and 40.
The sample size is greater than 40, without outliers.

This approach consists of four steps:

state the hypotheses
formulate an analysis plan
analyze sample data
interpret results.

1. State the hypotheses

The first step is to state the null hypothesis and an alternative hypothesis.

$$Null\ hypothesis(H_0): \mu r ifelse(two.sided,"=",ifelse(less,">=","<=")) r x$result$mu$$ $$Alternative\ hypothesis(H_1): \mu r ifelse(two.sided, "\\neq" ,ifelse(less,"<",">")) r x$result$mu$$

Note that these hypotheses constitute a r ifelse(two.sided,"two","one")-tailed test. r ifelse(two.sided,twoS,ifelse(less,lessS,greaterS)).

2. Formulate an analysis plan

For this analysis, the significance level is r (1-x$result$alpha)*100%. The test method is a one-sample t-test.

3. Analyze sample data.

Using sample data, we compute the standard error (SE), degrees of freedom (DF), and the t statistic test statistic (t).

$$SE = \frac{s}{\sqrt{n}} = \frac{r x$result$s}{\sqrt{r x$result$n}} = r round(x$result$se,2)$$ $$DF=n-1=r x$result$n-1=r round(x$result$DF,2)$$

$$t = (\bar{x} - \mu) / SE = (r x$result$m - r x$result$mu)/r round(x$result$se,2) = r round(x$result$t,3)$$

where s is the standard deviation of the sample, $\bar{x}$ is the sample mean, $\mu$ is the hypothesized population mean, and n is the sample size.

We can visualize the confidence interval of mean.

plot(x)

Since we have a r ifelse(two.sided,"two","one")-tailed test, the P-value is the probability that the t statistic having r round(x$result$DF,2) degrees of freedom is r if(!greater) "less than" r if(!greater) round(-abs(x$result$t),2) r if(!less) "or greater than " r if(!less) round(abs(x$result$t),2).

We use the t Distribution curve to find p value.

draw_t(DF=x$result$DF,t=x$result$t,alternative=x$result$alternative)

$$pt(r round(x$result$t,3),r x$result$DF) =r round(x$result$p,3) $$

4. Interpret results.

Since the P-value (r round(x$result$p,3)) is r ifelse(x$result$p>x$result$alpha,"greater","less") than the significance level (r x$result$alpha), we canr if(x$result$p>x$result$alpha) "not" reject the null hypothesis.

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/hypothesis-test/mean.aspx?tutorial=AP URL[Accessed Data: 1/23/2022].