<< Chapter < Page Chapter >> Page >

Calculators and computers can easily calculate any Student's-t probabilities. The TI-83,83+,84+ have a tcdf function to find the probability for given values of t. The grammar for the tcdf command is tcdf(lower bound, upper bound, degrees of freedom). However for confidence intervals, we need to use inverse probability to find the value of t when we know the probability.

A probability table for the Student's-t distribution can also be used. The table gives t-scores that correspond to the confidence level (column) and degrees of freedom (row). (The TI-86 does not have an invT program or command, so if you are using that calculator, you need to use a probability table for the Student's-t distribution.) When using t-table, note that some tables are formatted to show the confidence level in the column headings, while the column headings in some tables may show only corresponding area in one or both tails.

This is a picture of a t-table. A real t-table that is in table format and accessible is in the appendix of this book.

A Student's-t table (See the Table of Contents Tables ) gives t-scores given the degrees of freedom and the right-tailed probability. The table is very limited. Calculators and computers can easily calculate any Student's-t probabilities.

    The notation for the student's-t distribution is (using t as the random variable) is

  • T ~ t df where df = n - 1 .
  • For example, if we have a sample of size n=20 items, then we calculate the degrees of freedom as df=n−1=20−1=19 and we write the distribution as T ~ t 19

If the population standard deviation is not known , the margin of error for a population mean is:

  • ME = t α 2 ( s n )
  • t α 2 is the t-score with area to the right equal to α 2
  • use df = n - 1 degrees of freedom
  • s = sample standard deviation

The format for the confidence interval is:

( x - ME , x + ME ) .

Suppose you do a study of acupuncture to determine how effective it is in relieving pain. You measure sensory rates for 15 subjects withthe results given below. Use the sample data to construct a 95% confidence interval for the mean sensory rate for the population (assumed normal) fromwhich you took the data.

The solution is shown step-by-step and by using the TI-83, 83+ and 84+ calculators.

  • 8.6
  • 9.4
  • 7.9
  • 6.8
  • 8.3
  • 7.3
  • 9.2
  • 9.6
  • 8.7
  • 11.4
  • 10.3
  • 5.4
  • 8.1
  • 5.5
  • 6.9
  • Check the assumptions and conditions.
  • The first solution is step-by-step (Solution A).
  • The second solution uses an two column model(Solution B).

Solution a

Start by discussing the assumptions and conditions that support you model.
  • Randomization-The problem does not state that the sample was randomly selected. I will assume that it is in order to meet the conditions necessary to calculate a confidence interval.
  • Independence- The problem does not state that the 15 subjects are independent. I will assume that they are in order to meet the conditions necessary to calculate a confidence interval.
  • 10% condition-A sample size of 15 must be less than 10% of the total population receiving acupuncture to relieving pain. (A reasonable assumption).
  • Nearly Normal-The problem states that the population should be assumed to be nearly normal (skewed to the left in histogram but somewhat evenly distributed about the median). The histogram and boxplot below support that statement.

To find the confidence interval, you need the sample mean, x , and the ME.

x = 8.2267 s = 1.6722 n = 15

df = 15 - 1 = 14

CL = 0.95 so α = 1 - CL = 1 - 0.95 = 0.05

α 2 = 0.025 t α 2 = t .025

The area to the right of t .025 is 0.025 and the area to the left of t .025 is 1−0.025=0.975

t α 2 = t .025 = 2.14 using invT(.975,14) on the TI-84+ calculator.

ME = t α 2 ( s n )

ME = 2.14 ( 1.6722 15 ) = 0.924

x - ME = 8.2267 - 0.9240 = 7.3

x + ME = 8.2267 + 0.9240 = 9.15

The 95% confidence interval is (7.30, 9.15) .

We estimate with 95% confidence that the true population mean sensory rate is between 7.30 and 9.15.

When calculating the margin of error, a probability table for the Student's-t distribution can be used to find the value of t. The table gives t-scores that correspond to the confidence level (column) and degrees of freedom (row); the t-score is found where the row and column intersect in the table.

**With contributions from Roberta Bloom

Questions & Answers

Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
sciencedirect big data base
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
characteristics of micro business
for teaching engĺish at school how nano technology help us
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
what is the actual application of fullerenes nowadays?
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
is Bucky paper clear?
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Do you know which machine is used to that process?
how to fabricate graphene ink ?
for screen printed electrodes ?
What is lattice structure?
s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
On having this app for quite a bit time, Haven't realised there's a chat room in it.
what is biological synthesis of nanoparticles
Sanket Reply
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply
Practice Key Terms 7

Get the best Algebra and trigonometry course in your pocket!

Source:  OpenStax, Collaborative statistics using spreadsheets. OpenStax CNX. Jan 05, 2016 Download for free at http://legacy.cnx.org/content/col11521/1.23
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Collaborative statistics using spreadsheets' conversation and receive update notifications?