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Ch7.3 C.I.’s for Normal Distributions

This article explains the properties of t-distributions for normal distributions, including the bell-shaped curve, spread, and relationship with the standard normal curve. It also discusses t-critical values, confidence intervals, prediction intervals, and tolerance intervals.

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Ch7.3 C.I.’s for Normal Distributions

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  1. Ch7.3 C.I.’s for Normal Distributions When is the mean of a random sample of size n from a normal distribution with mean the RV has a probability distribution called a t distribution with n – 1 degrees of freedom (df). Properties of t Distributions Let tv denote the density function curve for vdf. • Each tv curve is bell-shaped and centered at 0. • Each tv curve is spread out more than the standard normal (z) curve Ch7.3

  2. 3. As v increases, the spread of the corresponding tv curve decreases. • As , the sequence of tv curves approaches the standard normal curve (the z curve is called a t curve with df = t Critical Value Let = the number on the measurement axis for which the area under the t curve with v df to the right of is called a tcritical value. Pictorial Definition of Ch7.3

  3. Confidence Interval Let and s be the sample mean and standard deviation computed from the results of a random sample from a normal population with mean The Ch7.3

  4. Prediction Interval A prediction interval (PI) for a single observation to be selected from a normal population distribution is The prediction level is Tolerance Interval Let k be a number between 0 and 100. A tolerance interval for capturing at least k% of the values in a normal population distribution with a confidence level of 95% has the form Ch7.3

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