Student's t Distribution:
Whenever the population standard deviation is not known, the mean consists of a Student's t-distribution. The Student's t distribution was made by an Irish brewery worker William T. Gosset. The brewery wouldn't permit him to publish his work beneath his name; therefore he used the pseudonym ‘Student’.
The Student's t-distribution is much similar to standard normal distribution.
a) It is symmetric on its mean.
b) It consists of a mean of zero.
c) It consists of a standard deviation and variance more than 1.
d) There are really many t-distributions, one for each degree of freedom.
e) Since the sample size rises, the t distribution approaches to the normal distribution.
f) This is bell shaped.
g) The t-scores can be positive or negative; however the probabilities are always positive.
Degrees of Freedom:
It is a degree of freedom which takes place for every data value that is permitted to differ once a statistic has been fixed. For single mean, there are (n-1) degrees of freedom. This value will modify depending on the statistic being employed.Population Standard Deviation Unknown:
When the population standard deviation, sigma is not known, then the mean consists of a student's t-distribution and the sample standard deviation is utilized rather than the population standard deviation.
The maximum error of estimate is provided by the formula for E as shown below. ‘t’ here is the t-score obtained from the Student's t-table. t-score is a factor of the level of confidence and sample size.
E = tα/2 (s/√n)
Once you have calculated E, now it is recommended to save it to the memory on your calculator. On TI-82, an excellent choice would be the letter E. The reason for this is that the limits for confidence interval are now found by adding up and subtracting the maximum error of estimate from or to sample mean.
Note that the formula is similar as for population mean when the population standard deviation is known. The only thing that has modified is the formula for maximum error of the estimate.
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