Question

Use three sentences to summarize 1. Confidence Interval on the Mean of a Normal Distribution, Variance Known , 2. Confidence Interval on the Mean of a Normal Distribution, Variance Unknown and 3. Tolerance and Prediction Intervals respectively.( can be an explanation or the main point of knowledge).

Answer 1

1.Confidence Interval on mean of a normal distribution,variance known:

For a population, whose mean is and standard deviation (given),confidence interval on sample mean based on the simple random sample of size n,is given by ( ),where Z* is the upper (1-C)/2 critical value for the standard normal distribution. An increase in sample size will decrease the length of the confidence interval. As the standard deviation decreases as n increases.

2.Confidence interval on mean of a normal distribution, variance unknown:

For a population whose mean is and standard deviation is not known so we estimate the standard deviation s. In this case the sample mean will no longer follows normal distribution. Instead it follows t distribution with mean and standard deviation. And for a sample size n,the degrees of freedom of t distribution is (n-1). The confidence interval is given by (),where t* is the upper (1-C)/2 critical value for t distribution with (n-1) degrees of freedom.

3.Tolerance and Prediction Intervals:

Tolerance Interval is a range that likely contains specific proportion of population.Here, we specially dealing with the spread of individual values around the mean. To create a tolerance interval,,we need to specify both the confidence level and the proportion.

After fitting a regression model, we can obtain prediction intervals. These intervals predict the value of the dependent variable given specific settings of the independent variable. A  confidence interval of the prediction is a range that likely contains the mean value of the dependent variable given specific values of the independent variable. These intervals provide a range for the population average.