# 8.2 A single population mean using the student t distribution  (Page 2/21)

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Calculators and computers can easily calculate any Student's t-probabilities. The TI-83,83+, and 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.

For the TI-84+ you can use the invT command on the DISTRibution menu. The invT command works similarly to the invnorm. The invT command requires two inputs: invT(area to the left, degrees of freedom) The output is the t-score that corresponds to the area we specified.

The TI-83 and 83+ do not have the invT command. (The TI-89 has an inverse T command.)

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 a 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.

A Student's t table (See [link] ) 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 (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 error bound for a population mean is:

• $EBM=\left({t}_{\frac{\alpha }{2}}\right)\left(\frac{s}{\sqrt{n}}\right)$ ,
• ${t}_{\frac{\sigma }{2}}$ is the t -score with area to the right equal to $\frac{\alpha }{2}$ ,
• use df = n – 1 degrees of freedom, and
• s = sample standard deviation.

The format for the confidence interval is:
$\left(\overline{x}-EBM,\overline{x}+EBM\right)$ .

To calculate the confidence interval directly:
Press STAT.
Arrow over to TESTS.
Arrow down to 8:TInterval and press ENTER (or just press 8).

Suppose you do a study of acupuncture to determine how effective it is in relieving pain. You measure sensory rates for 15 subjects with the results given. Use the sample data to construct a 95% confidence interval for the mean sensory rate for the population (assumed normal) from which you took the data.
The solution is shown step-by-step and by using the TI-83, 83+, or 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

Press STAT and arrow over to TESTS .
Arrow down to 8:TInterval and press ENTER (or you can just press 8 ).
Arrow to Data and press ENTER .
Arrow down to List and enter the list name where you put the data.
There should be a 1 after Freq .
Arrow down to C-level and enter 0.95
Arrow down to Calculate and press ENTER .
The 95% confidence interval is (7.3006, 9.1527)

## Note

When calculating the error bound, a probability table for the Student's t-distribution can also 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.

what percent of the students would be expected to score above 95?
inferential statistics is what?
in which we make infrences (hypothsis)
surpose a data set of 2,3,5,6,1,4 are given find median
lucy
Mean (average) 4... Median (middle term) 3.5.. Mode (frequency) every element in a set has 1 frequrncy
Akash
i arrange the data set in ascending order. that is, 1,2,3,4,5,6. then find the data set that falls in the middle. in this case, 3 & 4 fall in the middle. you then sum and obtain the average. that is, (3+4)/2=3.5. therefore, 3.5 is the median.
Gbenga
both of you are correct.
Joseph
hello guys
Abasikponke
thanks
lucy
great to be here
King
how does a line graph look
King
hi
Davia
hello
lucy
pls who knows how line graph look like
King
line graph usually have a straight line running through axis
Dike
am new here anyone willing to orient me?
Timothy
find the media of the following numbers 61,64,67,70,73
my body pls
lucy
67
Benmike
Benmike
what is the percentile for the set of data in the class C and frequency F(c,f)given by (9.3-9.7,2) (9.8-10.2,5) (10.3-10.7,12) (10.8-11.2,17) (11.3-11.7,14) (11.8-12.2,6) (12.3-12.7,3) (12.8-13.2,1)
how to find median
arrange ascending and desending order than the mid value is Median
rajendra
ok
Hrishe
what if it is a group data
Oloyede
mean/ medium/ mode
Michelle
n\2 and n+1\2
An operational manager at a manufacturing company is interested in the level of satisfaction of computer buyers. The manager has developed a satisfaction scale of 1-10 to mark their level of understanding with the company.What is the population of the interest?
Any clues
Virtual
how to use grouped and ungrouped data
Just a test from gplay
how come 5.67
by dividing 11.37 on 2
saifuddin
by dividing 11.34 on 2
saifuddin
what is index number?
vinayak
What is the differences between quota an lottery system of sampling
EGBE
What are the are the characteristics that are critically expedients in selecting the sample size
EGBE
fit a binomial distribution for the following data and test the goodness of fit x: 0 1 2 3 4 5 6 f: 5 18 28 12 7 6 4
solution
Mano
Mano
Simonsakala
It is a square chi
Nelson
But can't be a binomial because, the x numbers are 0 to 6, instead those would be "0" or "1" in a straight way
Nelson
You can do a chi-square test, but the assumption has to be a normal distribution, and the last f's number need to be "64"
Nelson
sorry the last f's numbers : "6 and 4" which are the observed values for 5 and 6 (expected values)
Nelson
hi
rajendra
can't understand basic of statistics ..
rajendra
Sorry I see my mistake, we have to calculate the expected values
Nelson
So we need this equation: P= (X=x)=(n to x) p^x(1-p)^n-x
Nelson
why it is not possible brother
ibrar
were n= 2 ( binomial) x= number of makes (0 to 6) and p= probability, could be 0.8.
Nelson
so after we calculate the expected values for each observed value (f) we do the chi-square. x^2=summatory(observed-expected)^2 / expected and compare with x^2 in table with 0.8
Nelson
tomorrow I'll post the answer, I'm so tired today, sorry for my mistake in the first messages.
Nelson
It is possible, sorry for my mistake
Nelson
two trader shared investment and buoght Cattle.Mr.Omer bought 255 cows & rented the farm for a period of 32 days. Mr. Ahmed grazed his Cattle for 25 days. Mr. Ahmed's cattle was 180 cows.Together they profited $7800. the rent of the farm is$ 3000 so divide the profit per gows/day for grazing day
Mohamed
how to start this book, who is reading thins first time
It is my first time reading this book
Good one
ihsan
from were did you get 2/50?m
People living longer
Why do you think that is?
Jazzy
because there is an increase in number of people with age more than 30.
Ok. And what do you think is the driving factor behind that hypothesis?
Jazzy
fewer birth and increase in # of years living or fewer dying
What about the improvement of technology and medicine?
Jazzy
godwin
technology and medicine is improving but they are limited
Ayunku
If those conscience of their health, one will live longer periods of life.
Montrae
hi,why the mean =sum(xi)/n but the variance =sum(xi-xbar)/ n-1 what is the difference between (n or n-1)
This is hard to type, so I'll use "m" for "x bar", and a few other notations that I hope will be clear: Definition: sqrt(SUM[(x - m)^2] / (n-1)) where m = SUM[x] / n Desired formula: sqrt((SUM[x^2] - SUM[x]^2)/n / (n-1)) Now let's do what you started to do, and see if we can manipulate the definitio
Michael
what is the difference between (n ) and (n-1) in the mean and variance
Soran
Definition: sqrt(SUM[(x - m)^2] / (n-1)) where m = SUM[x] / n what is the difference between (n and n-1)
Soran
Hi, the diference is tha when we estimate parameters in a sample (not in the total population) we need to consider the degrees of liberty for the estimation.
Nelson
Hie guys, am analysing rainfall data for different stations and i got kurtosis values of 0.7 for one station and 0.4 for another, what can i say about this?
Kudakwashe
hi
ujjal
difference in degrees of freedom
vinayak