**R**: use the functiont.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.

*Additional Notes:*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")"