Question

4.) When would you use the normal distribution? T-distribution? Chi-Square distribution?

Answer 1

1. When would you use normal distribution:-

Ans. The normal distribution is the most important probability distribution in statistics because it fits many natural phenomena. For example, heights, blood pressure, measurement error, and IQ scores follow the normal distribution. It is also known as the Gaussian distribution and the bell curve.

The normal distribution is a probability function that describes how the values of a variable are distributed. It is a symmetric distribution where most of the observations cluster around the central peak and the probabilities for values further away from the mean taper off equally in both directions.

You can use it to determine the proportion of the values that fall within a specified number of standard deviations from the mean. We use normal distribution to find proportion when population standard deviation ( ) is known. Z-tests are used in statistics to estimate significance. For example, in a normal distribution, 68% of the observations fall within +/- 1 standard deviation from the mean.

2. When would you use T - distribution:-

Ans. The T distribution, also known as the Student’s t-distribution, is a type of probability distribution that is similar to the normal distribution with its bell shape but has heavier tails. T distributions have a greater chance for extreme values than normal distributions, hence the fatter tails.

You can also use it to determine the proportion of the values that fall within a specified number of standard deviations from the mean. But, we use T-distribution to find proportion when population standard deviation ( ) is unknown. T-tests are used in statistics to estimate significance.

2. When would you use Chi-square distribution:-

Ans. A standard normal deviate is a random sample from the standard normal distribution. The Chi Square distribution is the distribution of the sum of squared standard normal deviates. The degrees of freedom of the distribution is equal to the number of standard normal deviates being summed. Therefore, Chi Square with one degree of freedom, written as is simply the distribution of a single normal deviate squared. The area of a Chi Square distribution below 4 is the same as the area of a standard normal distribution below 2, since 4 is 22 . The mean of chi square distribution is n and variance is 2n, where n is number of samples

Applications of Chi-square distribution:-

• Apllication in Genetics, where in breeding experiments it is it is desired to determine whether observed results are consistent with expected segregations.
• Application to contingency table, where it is used in testing independence in a contingency table.
• Application to fitting the distribution, it is used as test of goodness of fit of adistribution.
• To test the variance of a normal population.
• To test the homogeniety of independent estimates of population variances.