Question

What is a t-distribution? Compare the t-distribution and the standard normal distribution. When do you use the t-distribution? The height of US women is normally distributed. A sample of 25 women shows an average hight of 160cm with a standard deviation of 5cm. Use t distribution to find the probability that the average height of randomly selected 25 women is greater than 172cm.

Answer 1

When you have a normal distribution as a population but you do not know its mean or standard deviation, in this situation the t-distribution is used to determine the mean.

So basically a t-distribution is derived by taking samples from a normal distribution whose parameters are not known to you.

The t-distribution resembles the normal distribution, but it is a bit lower and wider at the extreme ends. When you increase the sample size, then the resulting t-distribution approaches the standard normal distribution.

Data given is:

Sample size n = 25

Sample mean m = 160

Sample standard deviation S = 5

So standard error, S' = S/(n^0.5) = 5/(25^0.5) = 1

Now, at X = 172, we calculate the test statistic as follows:

t = (X-m)/S' = (172-160)/1 = 12

Degrees of freedom df = n-1 = 24

So the corresponding p-value for this t-value is:

P(X > 172) = P(t > 4.8 with df = 24) = less than 0.00001