## Tech Tips: One-Sample Means Test

### One-Sample Means Test ($\sigma$ unknown)

To conduct a $t$-test that a population's mean is $\mu$ given sample data and a confidence level, but with no knowledge of the population standard deviation $\sigma$,
• R: use the function

t.test(data,alternative,mu,conf.level)

To explain the parameters:

• data is a vector consisting of the sample data
• mu is the mean $\mu$ associated with the null hypothesis
• alternative is a string of text that specifies the alternative hypothesis (i.e., "two.sided", "less", or "greater")

Consider the following example of this function's use:

Suppose the weights (in grams) of a sample of eleven small screws are found to be $$0.38,0.55,1.54,1.55,0.50,0.60,0.92,0.96,1.00,0.86,1.46$$ The production process for the screws is supposed to result in screws with mean weight of $1$ gram. Assuming the weights are normally distributed, test this claim at a $0.10$ significance level.

> data = c(0.38,0.55,1.54,1.55,0.50,0.60,0.92,0.96,1.00,0.86,1.46)
> t.test(data,alternative="two.sided",mu=1.00,conf.level=0.90)

One Sample t-test

data:  data
t = -0.48485, df = 10, p-value = 0.6382
alternative hypothesis: true mean is not equal to 1
90 percent confidence interval:
0.7070946 1.1692691
sample estimates:
mean of x
0.9381818

Given the $p$-value given above, which is greater than the significance level, this sample does not provide any statistically significant evidence that the mean weight is not $1$ g.

If all one wishes to calculate is the confidence interval for a population mean given a sample taken from it -- one can simply pass to t.test() the data and conf.level arguments and look at the conf.int component of the resulting list, as seen below.

> data = c(68,73,68,70,75,57,64,67,74,64,64,66,71,66,59,66)
> t.test(data,conf.level=0.95)$conf.int [1] 64.35351 69.64649 attr(,"conf.level") [1] 0.95 When conducting a one-tailed test, one should use alternative="less" or alternative="greater", as appropriate. If one should desire to store the$p$-value in a variable to use for some other purpose, one can extract it from the overall test results in the following way: > test.results = t.test(data,alternative="two.sided",mu=1.00,conf.level=0.90) > test.results$p.value
[1] 0.6382267

Similarly, we can retrieve the upper and lower bounds of the related confidence interval with

> test.results = t.test(data,alternative="two.sided",mu=1.00,conf.level=0.90)
> test.results$conf.int[c(1,2)] [1] 0.7070946 1.1692691 • Excel: One can build a worksheet for conducting a one sample test concerning a mean when the population's standard deviation is unknown using the functions related to a$t\$-distribution. Below is an example:

Here are the relevant formulas:

F8:"=COUNTA(C:C)"                              # the COUNTA() function counts non-empty
F9:"=AVERAGE(C:C)"                             # cells in the range given to it
F10:"=STDEV.S(C:C)"
F11:"=F8-1"
F13:"=(F9-F4)/(F10/SQRT(COUNTA(C:C)))"

F14:"=IF(EXACT(TRIM(F5),"two.sided"),          # the TRIM() function removes extra spaces
T.INV(F6/2,F11),
IF(EXACT(TRIM(F5),"less"),            # the EXACT() function returns TRUE when
T.INV(F6,F11),                     # the two strings passed to it agree, and
IF(EXACT(TRIM(F5),"greater"),      # FALSE otherwise
T.INV(1-F6,F11),
"ERROR")))"                     # the IF(condition,a,b) function returns
# a when condition is TRUE, b otherwise
F15:"=IF(EXACT(TRIM(F5),"two.sided"),
2*(1-T.DIST(ABS(F13),F11,TRUE)),
IF(EXACT(TRIM(F5),"less"),
T.DIST(F13,F11,TRUE),
IF(EXACT(TRIM(F5),"greater"),
1-T.DIST(F13,F11,TRUE),
"ERROR")))"

F17:"=IF(F15<F6,"REJECT NULL HYPOTHESIS","FAIL TO REJECT NULL HYPOTHESIS")"