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In practice, we rarely know the population standard deviation . In the past, when the sample size was large, this did not present a problem to statisticians. They used the sample standard deviation s as an estimate for σ and proceeded as before to calculate a confidence interval with close enough results. However, statisticians ran into problems when the sample size was small. A small sample size caused inaccuracies in the confidence interval.

William S. Goset (1876–1937) of the Guinness brewery in Dublin, Ireland ran into this problem. His experiments with hops and barley produced very few samples. Just replacing σ with s did not produce accurate results when he tried to calculate a confidence interval. He realized that he could not use a normal distribution for the calculation; he found that the actual distribution depends on the sample size. This problem led him to "discover" what is called the Student's t-distribution . The name comes from the fact that Gosset wrote under the pen name "Student."

Up until the mid-1970s, some statisticians used the normal distribution approximation for large sample sizes and only used the Student's t-distribution only for sample sizes of at most 30. With graphing calculators and computers, the practice now is to use the Student's t-distribution whenever s is used as an estimate for σ .

If you draw a simple random sample of size n from a population that has an approximately a normal distribution with mean μ and unknown population standard deviation σ and calculate the t -score t = x ¯ μ ( s n ) , then the t -scores follow a Student's t-distribution with n – 1 degrees of freedom . The t -score has the same interpretation as the z -score . It measures how far x ¯ is from its mean μ . For each sample size n , there is a different Student's t-distribution.

The degrees of freedom , n – 1 , come from the calculation of the sample standard deviation s . In [link] , we used n deviations ( x x ¯ values ) to calculate s . Because the sum of the deviations is zero, we can find the last deviation once we know the other n – 1 deviations. The other n – 1 deviations can change or vary freely. We call the number n – 1 the degrees of freedom (df).

    Properties of the student's t-distribution

  • The graph for the Student's t-distribution is similar to the standard normal curve.
  • The mean for the Student's t-distribution is zero and the distribution is symmetric about zero.
  • The Student's t-distribution has more probability in its tails than the standard normal distribution because the spread of the t-distribution is greater than the spread of the standard normal. So the graph of the Student's t-distribution will be thicker in the tails and shorter in the center than the graph of the standard normal distribution.
  • The exact shape of the Student's t-distribution depends on the degrees of freedom. As the degrees of freedom increases, the graph of Student's t-distribution becomes more like the graph of the standard normal distribution.
  • The underlying population of individual observations is assumed to be normally distributed with unknown population mean μ and unknown population standard deviation σ . The size of the underlying population is generally not relevant unless it is very small. If it is bell shaped (normal) then the assumption is met and doesn't need discussion. Random sampling is assumed, but that is a completely separate assumption from normality.

Questions & Answers

How to answer quantitative data
Alhassan Reply
what's up here ... am new here
sorry question a bit unclear...do you mean how do you analyze quantitative data? If yes, it depends on the specific question(s) you set in the beginning as well as on the data you collected. So the method of data analysis will be dependent on the data collecter and questions asked.
how to solve for degree of freedom
Quantitative data is the data in numeric form. For eg: Income of persons asked is 10,000. This data is quantitative data on the other hand data collected for either make or female is qualitative data.
Degree of freedom is the unconditionality. For example if you have total number of observations n, and you have to calculate variance, obviously you will need mean for that. Here mean is a condition, without which you cannot calculate variance. Therefore degree of freedom for variance will be n-1.
data that is best presented in categories like haircolor, food taste (good, bad, fair, terrible) constitutes qualitative data
vegetation types (grasslands, forests etc) qualitative data
I don't understand how you solved it can you teach me
Caleb Reply
solve what?
What is the end points of a confidence interval called?
lower and upper endpoints
Class members write down the average time (in hours, to the nearest half-hour) they sleep per night.
William Reply
how we make a classes of this(170.3,173.9,171.3,182.3,177.3,178.3,174.175.3)
why is always lower class bundry used
Assume you are in a class where quizzes are 20% of your grade, homework is 20%, exam _1 is 15%,exam _2 is 15%, and the final exam is 20%.Suppose you are in the fifth week and you just found out that you scored a 58/63 on the fist exam. You also know that you received 6/9,8/10,9/9 on the first
Diamatu Reply
quizzes as well as a 9/11,10/10,and 4.5/7 on the first three homework assignment. what is your current grade in the course?
the answer is 2.6
if putting y=3x examine that correlation coefficient between x and y=3x is 1.
Aadrsh Reply
what is permutation
Rodlett Reply
how to construct a histogram
Baalisi Reply
You have to plot the class midpoint and the frequency
ok so you use those two to draw the histogram right.
ok can i be a friend so you can be teaching me small small
how do you calculate cost effectiveness?
Hi everyone, this is a very good statistical group and am glad to be part of it. I'm just not sure how did I end up here cos this discussion just popes on my screen so if I wanna ask something in the future, how will I find you?
To make a histogram, follow these steps: On the vertical axis, place frequencies. Label this axis "Frequency". On the horizontal axis, place the lower value of each interval. ... Draw a bar extending from the lower value of each interval to the lower value of the next interval.
I really appreciate that
umar Reply
I want to test linear regression data such as maintenance fees vs house size. Can I use R square, F test to test the relationship? Is the good condition of R square greater than 0.5
Mok Reply
yes of course must have use f test and also use t test individually multple coefficients
hi frnd I'm akeem by name, I wanna study economics and statistics wat ar d thing I must do to b a great economist
Is R square cannot analysis linear regression of X vs Y relationship?
To be an economist you have to be professional in maths
hi frnds
what is random sampling what is sample error
Nistha Reply
@Nistha Kashyap Random sampling is the selection of random items (or random numbers) from the group. A sample error occurs when the selected samples do not truely represent the whole group. The can happen when most or all of the selected samples are taken from only one section of the group;
Thus the sample is not truely random.
What is zero sum game?
Hassan Reply
A game in which there is no profit & no loss to any of the both player.
Differences between sample mean & population mean
mohammed Reply
Not difference in the formula except the notation, sample mean is denoted by x bar and population mean is denoted by mu symbol. There is formula as well as notation between difference variance and standard deviations
Likely the difference would be in the result, unless the sample is an exact representation of the population (which is unlikely.)
what is data
Nii Avin - Data is just a simple way to refer to the numbers in the population, or in the sample used in your calculations.
what are the types of data
Data is the very pale android from the Star Trek Enterprise
Am Emmanuel from Nigeria
Am Qudus from Nigeria
am Handson from Cameroon
what is a mode?
Nii - data is whatever you are sampling. Such as the number of students in each classroom.
Handson Ndintek - the mode is the number appearing most frequently. Example: 7 9 11 7 4 6 3 7 2. 7 is the mode. In a group such as 7 9 1 4 6 3, there is no mode because no number appears more often than any other.
hi I want to know how to find class boundary
give me the two types of data
Neddy Reply
qualitative and quantitative
primary and secondary data
qualitative and quantitative
Using Cauchy Schwartz inequality,or prove that b2-b1-1=0
Md Reply
what is the ongoing probability that President Trump will remain in the position he has chosen as his viability of his cabinet as he runs for reelection in the primaries of 2020 election year

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Source:  OpenStax, Introductory statistics. OpenStax CNX. May 06, 2016 Download for free at http://legacy.cnx.org/content/col11562/1.18
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