July 9, 2012- Confidence Intervals for the Mean (Standard Deviation is Known)

            

The III- Gold students are again facing a new challenge. A challenge to master the new lesson which our teacher taught us on July 9, 2012 which is the confidence intervals for the mean when the standard deviation is given.

According to our teacher:

Normal Distribution can only be used to find confidence intervals for the mean when the variable is normally distributed and the population standard distribution is known.

If one desires to be more confident the interval must be larger but statisticians prefer a short interval with high degree of confidence.

Formula to find the Confidence Intervals for the Mean

 

To find Z a_/2.

Example 1.

The principal of a high school wishes to estimate  the average height of students presently enrolled. From past studies, the standard deviation is known to be 3 years. A sample of students is selected, and the mean is found to be 24.3. Find the 90% confidence interval population mean.

Solution.

Additional Fact about the Topic:

In statistics, a confidence interval is a kind of interval estimate of a population parameter and is used to indicate the reliability of an estimate. It is an observed interval, in principle different from sample to sample, that frequently includes the parameter of interest, if the experiment is repeated. How frequently the observed interval contains the parameter is determined by the confidence level or confidence coefficient. More specifically, the meaning of the term "confidence level" is that, if confidence intervals are constructed across many separate data analyses of repeated experiments, the proportion of such intervals that contain the true value of the parameter will match the confidence level; this is guaranteed by the reasoning underlying the construction of confidence intervals

The value z^* representing the point on the standard normal density curve such that the probability of observing a value greater than z^* is equal to pis known as the upper p critical value of the standard normal distribution. For example, if p = 0.025, the value z^* such that P(Z > z^*) = 0.025, or P(Z < z^*) = 0.975, is equal to 1.96. For a confidence interval with level C, the value p is equal to (1-C)/2. A 95% confidence interval for the standard normal distribution, then, is the interval (-1.96, 1.96), since 95% of the area under the curve falls within this interval.

BY: CARL JOEL E. PALMA III-GOLD