<< 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

where we get a research paper on Nano chemistry....?
Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
ya I also want to know the raman spectra
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
yes that's correct
I think
Nasa has use it in the 60's, copper as water purification in the moon travel.
nanocopper obvius
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
how nano science is used for hydrophobicity
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
what is differents between GO and RGO?
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
analytical skills graphene is prepared to kill any type viruses .
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
The nanotechnology is as new science, to scale nanometric
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
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.
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
Privacy Information Security Software Version 1.1a
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply
Practice Key Terms 7

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play

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?