To check Validity Conditions for Theory-Based methods for Inference with One Mean
To apply Theory-Based methods for Inference with One Mean and draw appropriate conclusions.
To apply calculation techniques using tools from Lab 1.
To learn to display quantitative data in a histogram.
As usual, we start by loading our two packages: mosaic
and ggformula
. To load a package, you use the
library()
function, wrapped around the name of a package.
I’ve put the code to load one package into the chunk below. Add the
other package you need.
We’ll load the example data, GSS_clean.csv
. It is
available at this Url: https://raw.githubusercontent.com/IJohnson-math/Math138/main/GSS_clean.csv.
We’ll use the read.csv()
function to read in the data.
#load data
GSS <- read.csv("https://raw.githubusercontent.com/IJohnson-math/Math138/main/GSS_clean.csv")
This dataset comes from the General Social Survey (GSS), which is collected by NORC at the University of Chicago. It is a random sample of households from the United States, and has been running since 1972, so it is very useful for studying trends in American life. The data I’ve given you is a subset of the questions asked in the survey, and the data has been cleaned to make it easier to use. But, there are still some messy aspects (which we’ll discover as we analyze it further throughout this class!).
Suppose we wanted to know for all U.S. workers if the mean number of hours worked in a week is different than 40. We could write our null and alternative hypotheses as
\[ H_0: \mu = 40 \\ H_a: \mu\neq 40 \]
In Lab 1, we used the following commands to view parts of the GSS
data: glimpse
, head
, tail
. You
can also view the data in another tab by clicking on ‘GSS’ in the
Environment pane. This allows you to scroll up and
down, and left and right to view the data.
You may use these commands on the GSS data to review the data.
In this lab we will consider the mean of
number_of_hours_worked_last_week
as our statistic of
interest. Looking within the GSS data we see many NA values for the
variable number_of_hours_worked_last_week
. Let’s start by
filtering out the NA values. The command filter
is used to
keep the observational units that satisfy a given property. In this
example the property is
!is.na(number_of_hours_worked_last_week)
; here, the
exclamation point, !
, is read as “not”, so this command
keeps the observational units that do not have NA as an entry
for the variable number_of_hours_worked_last_week
.
Let’s look at the data. Create a histogram of
number_of_hours_worked_last_week
.
gf_histogram(~number_of_hours_worked_last_week, data=GSS, binwidth=4, title="Time Worked Last Week by a Random Sample of 1381 US Adults", xlab="Time Worked (hours)")
Now we can compute the mean, the value of our point estimate. We name
the statistic xbar
(in place of the symbol \(\bar{x}\)).
## [1] 41.28168
The quantitative variable should have a symmetric distribution, or you should have at least 20 observations and the sample distribution should not be strongly skewed.
When these conditions are met we can use the \(t\)-distribution to approximate the \(p\)-value for our hypothesis test. It’s important to keep in mind that these conditions are rough guidelines and not a guarantee. All theory-based methods are approximations. They will work best when the sample distribution is symmetric, the sample size is large, and there are no large outliers. When in doubt, use a simulation-based method as a cross-check.
Check Validity Conditions: In this example we have \(n=1381\) observations, which is much larger than 20, and our sample distribution is symmetric as seen above in the histogram. Thus the validity conditions for theory-based inference with one mean are satisfied.
The standardized statistic, \(t\), is found using the formula \[ t = \frac{\bar{x} - \mu}{SE(\bar{x})} \]
and standard error for the null distribution is given by
\[ SE(\bar{x})=\frac{s}{\sqrt{n}}. \]
Calculate the standardized \(t\)- statistic
#calculate the standard deviation of the sample, s
s <- sd( ~number_of_hours_worked_last_week, data=GSS)
# n is the number of observational units (after filtering)
n=1381
#calculate standard error
SE <- s/sqrt(n)
#mu is the mean of the null hypotheses
mu = 40
#now we can calculate (and display) the standardized statistic
t <- (xbar - mu)/SE
t
## [1] 3.289241
We use the command t.test
to calculate a \(p\)-value. As we saw in Lab 1, the options
for alternative are “two.sided”, “greater”, “less” depending on the
inequality in the alternative hypotheses. We must also enter the
null-hypothesis parameter mu
(in place of the symbol \(\mu\)).
##
## One Sample t-test
##
## data: number_of_hours_worked_last_week
## t = 3.2892, df = 1380, p-value = 0.00103
## alternative hypothesis: true mean is not equal to 40
## 95 percent confidence interval:
## 40.51729 42.04607
## sample estimates:
## mean of x
## 41.28168
Our data is from a random sample of \(n=1381\) US workers collected through the General Social Survey. Since our sample is random, we may generalize our findings to the larger population of US workers. We consider the number of hours worked last week, a quantitative variable, and investigated whether or not the mean number of hours worked last week by US workers is equal to 40.
What can be concluded from the \(t\)-statistic and \(p\)-value?
Our statistic, the sample mean of \(\bar{x}=41.28\) hours worked last week, is 3.29 standard deviations away from the hypothesized mean of 40 hours worked last week. An observed statistic that is more than 3 standard deviations away from the hypothesized mean, as our is here, is very strong evidence against the null hypothesis. We are very unlikely to obtain a random sample of \(n=1381\) people with sample mean of \(\bar{x}\)=41.28 hours worked last week if the true population of US workers worked an average of 40 hours last week.
Similarly, the \(p\)-value of 0.00103 is very small. When a \(p\)-value is less than 0.01, as ours is here, we have very strong evidence against the null hypothesis. Thus, we will reject the null hypothesis (as it is not plausible) and accept the alternative hypothesis that mean number of hours worked by US workers is not equal to 40 hours per week.
Notice that the \(t\)-statistic and \(p\)-value give the same conclusions, as expected.