Hypothesis test for a difference between means

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(n1=100,n2=100,m1=200,s1=40,m2=190,s2=20,mu=7,alpha=0.05,alternative="greater")
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 difference between sample means is too big or if it is too small."
lessS="The null hypothesis will be rejected if the difference between sample means is too small."
greaterS="The null hypothesis will be rejected if the difference between sample means 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")

library(glue)
library(interpretCI)
if(two.sided){
if(x$result$mu==0) {
     claim= "Test the hypothesis that men and women spend equally on refreshment"
} else {
     claim= paste0("The team owner claims that the difference in spending money between men and women is equal to ", x$result$mu) 
}
} else{
    if(x$result$mu==0) {
     claim= paste0("Test the hypothesis that men spend", ifelse(greater,"more","less")," than women on refreshment")
   } else {
       claim= paste0("The team owner claims that men spend at least $", x$result$mu, 
                   ifelse(greater," more"," less")," than women") 
   }
}

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

string=glue("The local baseball team conducts a study to find the amount spent on refreshments at the ball park. Over the course of the season they gather simple random samples of {x$result$n1} men and {x$result$n2} women. For men, the average expenditure was ${x$result$m1}, with a standard deviation of ${x$result$s1}. For women, it was ${x$result$m2}, with a standard deviation of ${x$result$s2}.

{claim}. Assume that the two populations are independent and normally distributed.")

textBox(string)

Hypothesis test

This lesson explains how to conduct a hypothesis test for the difference between two means. The test procedure, called the two-sample t-test, is appropriate when the following conditions are met:

• The sampling method for each sample is simple random sampling.

• The samples are independent.

• Each population is at least 20 times larger than its respective sample.

• The sampling distribution is approximately normal, which is generally the case if any of the following conditions apply.

  • The population distribution is normal.

  • The population data are symmetric, unimodal, without outliers, and the sample size is 15 or less.

  • The population data are slightly skewed, unimodal, without outliers, and the sample size is 16 to 40.

  • The sample size is greater than 40, without outliers.

This approach consists of four steps:

1. state the hypotheses

2. formulate an analysis plan

3. analyze sample data

4. interpret results.

1. State the hypotheses

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

$$Null\ hypothesis(H_0): \mu_1-\mu_2 r ifelse(two.sided,"=",ifelse(less,">=","<=")) r x$result$mu$$ $$Alternative\ hypothesis(H_1): \mu_1-\mu_2 r ifelse(two.sided, "\\neq" ,ifelse(less,"<",">")) r x$result$mu$$

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%. Using sample data, we will conduct a two-sample t-test of the null hypothesis.

3. Analyze sample data

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

$$SE=\sqrt{\frac{s^2_1}{n_1}+\frac{s^2_2}{n_2}}$$ $$SE=\sqrt{\frac{r x$result$s1^2}{r x$result$n1}+\frac{r x$result$s2^2}{r x$result$n2}}$$ $$SE=r round(x$result$se,3)$$

$$DF=\frac{(\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2})^2}{\frac{(s_1^2/n_1)^2}{n_1-1}+\frac{(s_2^2/n_2)^2}{n_2-1}}$$

$$DF=\frac{(\frac{r x$result$s1^2}{r x$result$n1}+\frac{r x$result$s2^2}{r x$result$n2})^2}{\frac{(r x$result$s1^2/r x$result$n1)^2}{r x$result$n1-1}+\frac{(r x$result$s2^2/r x$result$n2)^2}{r x$result$n2-1}}$$ $$DF=r round(x$result$DF,2)$$

$$t=\frac{(\bar{x_1}-\bar{x_2})-d}{SE} = \frac{(r x$result$m1 -r x$result$m2)-r x$result$mu}{r round(x$result$se,3)}=r round(x$result$t,3)$$

where $s_1$ is the standard deviation of sample 1, $s_2$ is the standard deviation of sample 2, $n_1$ is the size of sample 1, $n_2$ is the size of sample 2, $\bar{x_1}$ is the mean of sample 1, $\bar{x_2}$ is the mean of sample 2, d is the hypothesized difference between population means, and SE is the standard error.

We can plot the mean difference.

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=round(x$result$DF,2),t=x$result$t,alternative=x$result$alternative)

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/difference-in-means.aspx?tutorial=AP URL[Accessed Data: 1/23/2022].

interpretCI documentation built on Jan. 28, 2022, 9:07 a.m.