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

The t-Distribution is the probability distribution that is used to estimate the population mean when the sample size is small (usually smaller than 30) and the population standard deviation is also unknown.

According to the central limit theorem, the sampling distribution of a statistic (like a sample mean) will follow anormal distribution, as long as the sample size is sufficiently large. Therefore, when we know the standard deviation of the population, we can compute a z-score, and use the normal distribution to evaluate probabilities with the sample mean.

t- distribution Vs Standard normal distribution:

The z-distribution assumes that the population standard deviation is known.
The t-distribution is based on the assumption that population standard deviation is not known and the sample standard deviation is an estimate of the population standard deviation. Both are used for estimating population parameters.

Let X denote the height of US women.

Required probability =