# CONFIDENCE INTERVAL ESTIMATES OF A POPULATION MEAN Stats Assignment Homework Help

CONFIDENCE INTERVAL ESTIMATES OF A POPULATION MEAN

Meaning of the Confidence Interval Estimate An interval estimate of IL is an interval of values a to b within which an unknown Population mean is  expected to lie. The interval is an inference based upon (1) the value of the mean x of a simple random sample selected from the population, and (2) known facts about the-sampling distribution of the mean (Chapter 9). We cannot be 100 percent certain that such an in (and usually a very small I’ rt) of the population. A confine. mat. of p. is an interval time toll there with a statement of how confident (e.I” 90 percent, 95 percent) we are the t the interval  correct.   the confide NE interval concept. The choice of m tho u in con instruct confide int eval for p. led and pin whether or not the population i normal and what are the population standard devotion (1. i known or unknown.

Confluence. .TLC. of, Norm.1 Population, (Tar Known popup e we have a normal population who mean and st standard deviation arc pad and (Then (from Chapter 9) the s dumpling distribution of the mean will be normal with a mean of p. and a standard error of For the sampling distribution of the mean, the standard normal variable is sample means X3, X., and so on. The question in each case is “Is J.L in this interval?”
The answer is that J.L will be in 95 percent of the intervals, and J.L will
not be in 5 percent of the intervals, as illustrated in Figure 10.6.
Keeping the last statement in mind, suppose we select only one sample of
size n from a population, compute the sample mean X, then state that J.L is
between X – 1.96<1’\$ and X + 1.96<1′.r. The statement is either correct (p. is in the interval), or incorrect (J.L is not in the interval). However, we are quite confident that the statement is correct because 95 percent of such statements are correct. The proportion of correct statements (0.95 in our illustration) is called the confidence coefficient C. The number 100C percent (95 percent in our illustration) is called the confidence level. The proportion of incorrect statements is symbolized by the Greek letter a (alpha). The sum of the proportions of correct and incorrect statements is 1; so.

C+a=1 or a = 1 – C

We can describe C as the chance that a confidence interval is correct, and a
as the  chance that an interval is incorrect. ! – 1.960″, S IJ. S ! + 1.960″

so – 1. 96(2) s IJ. s SO + 1. 96(2)
46.1 s IJ. s 53.9

EXERCISE Shafts made by an automatic machine are normally distributed
and have a standard deviation of 0.02 mm (millimeter); A sample of four shafts has a mean diameter of 25.01 mm. Construct a 95 percent confidence interval estimate of the mean diameter of all shafts made by the machine.
ANSWER 24.99 rnrn es p. :s 25.03 The multiplier 1.96 we have been using is the % value that corresponds to a confidence coefficient of C = 0.95. It arose because we started with an interval centered at IJ.which included 95 percent of the area under the normal curve (see Figure 10.3). Specifically, 1.96 is the % value which gives an area of0.95/2 = 0.4750 on each side of the mean; that is, from Table V(%IP = 0.4750) = 1.96 The procedure for obtaining the z value corresponding to a stated confidence coefficient can be simplified by using a table which gives the value of zfor a specified one-tail area. Suppose, for example, that we want C to be0.99. Then, as shown.in Figure 10;7, 0.99 is the central area, and the area in the two tails is Posted on August 27, 2014 in STATISTICAL INFERENCE ESTIMATION
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