Know T — Distribution and Z — Test
The Z distribution is a special case of the normal distribution with a mean of 0 and standard deviation of 1. The t-distribution is similar to the Z-distribution, but is sensitive to sample size and is used for small or moderate samples when the population standard deviation is unknown.
Let us discuss both one by one simultaneously.
T — distribution: -
The t-distribution, also known as Student’s t-distribution, is a way of describing data that follow a bell curve when plotted on a graph, with the greatest number of observations close to the mean and fewer observations in the tails.
It is a type of normal distribution used for smaller sample sizes, where the variance in the data is unknown.
T-distribution and T-score: -
A t-score is the number of standard deviation from the mean in a t-distribution. You can typically look up a t-score in at-table, or by using an online t-score calculator.
In statistics, t-scores are primarily used to find two things:
· The upper and lower bounds of a confidence interval when the data are approximately normally distributed.
· The p-value of the test statistic for t-tests and regression tests.
What is the difference between T— distribution and standard normal distribution?
The t-distribution gives more probability to observations in the tails of the distribution than the standard normal distribution (a.k.a. the z-distribution).
In this way, the t-distribution is more conservative than the standard normal distribution: to reach the same level of confidence or statistical significance, you will need to include a wider range of the data.
Properties of the T-distribution: -
1. The variable in t-distribution ranges from -∞ to +∞ (-∞ < t < +∞).
2. t- distribution will be symmetric like the normal distribution if the power of t is even in the probability density function(pdf).
3. For large values of ν(i.e., increased sample size n); the t-distribution tends to a standard normal distribution. This implies that for different ν values, the shape of t-distribution also differs.
4. The t-distribution is less peaked than the normal distribution at the centre and higher peaked in the tails. From the above diagram, one can observe that the red and green curves are less peaked at the centre but higher peaked at the tails than the blue curve.
5. The value of y(peak height) attains highest at μ = 0 as one can observe the same in the above diagram.
6. The mean of the distribution is equal to 0 for ν > 1 where ν = degrees of freedom, otherwise undefined.
7. The median and mode of the distribution is equal to 0.
Z — test: -
A z-test is a statistical test used to determine whether two population means are different when the variances are known and the sample size is large. A z-test is a hypothesis test in which the z-statistic follows a normal distribution.
A z-statistic, or z-score, is a number representing the result from the z-test. Z-tests are closely related to t-tests, but t-tests are best performed when an experiment has a small sample size.
Z-tests assume the standard deviation is known, while t-tests assume it is unknown.
Note: — The z-test is best used for greater-than or equal to 30 samples (n ≥ 30)because, under the central limit theorem, as the number of samples gets larger, the samples are considered to be approximately normally distributed.
When should you use a Z-test?
If the standard deviation of the population is unknown and the sample size is greater than or equal to 30, then the assumption of the sample variance equaling the population variance should be made using the z-test.
Regardless of the sample size, if the population standard deviation for a variable remains unknown, a t-test should be used instead.
What is a Z-score?
A z-score, or z-statistic, is a number representing how many standard deviations above or below the mean population the score derived from a z-test is.
Essentially, it is a numerical measurement that describes a value’s relationship to the mean of a group of values.
If a z-score is 0, it indicates that the data point’s score is identical to the mean score. A z-score of 1.0 would indicate a value that is one standard deviation from the mean.
Note: — Z-scores may be positive or negative, with a positive value indicating the score is above the mean and a negative score indicating it is below the mean.
What’s the difference between a T-test and Z-test?
Z-tests are closely related to t-tests, but t-tests are best performed when the data consists of a small sample size, i.e., less than 30.
Also, t-tests assume the standard deviation is unknown, while z-tests assume it is known.
Conclusion: -
In conclusion, the t-distribution and the z-test are important statistical concepts used for hypothesis testing and confidence interval estimation.
Understanding the t-distribution and z-test enables researchers and analysts to appropriately select and apply the correct statistical methods for hypothesis testing and confidence interval estimation. It allows for accurate inference and decision-making based on the nature of the data and the available sample size and information.
