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Statistics : T-Distribution and Z-Distribution

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Please help with the following problems. Provide step by step calculations and diagrams.

a. What is the difference in appearance between a t-distribution and a z-distribution? Why do they have slightly different appearances?

b. When do we use a t-distribution rather than a z-distribution?

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a. t- distribution is a probability distribution that is often used when the population standard deviation is not known or when the sample size is small. t- distribution is symmetrical, bell-shaped, and similar to the standard normal curve. It differs from the standard z-distribution curve, however, in that it has an additional parameter, called degrees of freedom, which changes its shape.
Figure 1. t-distribution ...

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