confidence intervals by stariya


       Read the following mini-lesson and follow the examples provided. Do not turn anything in
       to me. On the Final Exam, two optional questions will be provided on this topic. If you get
       these questions correct, I will replace your lowest homework or quiz score with a perfect

Motivation (Section 8.1)

       All semester we have assumed that we know or can calculate the mean and variance of a
       population. However, we usually do not know the true population mean or standard deviation
       for a random variable. For example, we might say that X = the weight of a milk container, and
       assume that X has a normal distribution with mean 2.012 liters and variance 0.23… but do we
       really know the average weight of ALL milk containers? Obviously not. These numbers are
       really just our best guess.

       This “best guess” comes from collecting data. We would collect milk cartons, calculate the
       mean and variance of the entire sample, and use these numbers as our estimates. In statistics,
       we use confidence intervals to determine how accurate these estimates are. In MTH 352, we
       discuss confidence intervals in great detail. However, for now, we only need to know how
       they are calculated.


       To estimate the parameters for the milk carton example, suppose we collect 50 milk cartons
       and weigh them. The dataset of milk weights is given on page 269 of your textbook. It turns
       out that the average (x ) and standard deviation ( s ) of this data set are as follows:
       x = 2.0727 liters
        s = 0.0711 liters

       In section 8.1, we learn how to use the t-distribution to build a confidence interval aroundx.
       This will give us an idea of our accuracy level. The required equation is in the blue box on
       page 328. This calculation involves five numbers: x, s, n (the sample size), a t-statistic, and
       . Think of  as being the “leftover probability”. So for a 90% confidence interval,  = 10%
       (i.e. 0.10); for a 95% confidence interval,  = 5% (i.e., 0.05); etc.

       Lastly, keep in mind that to calculate the t-statistic, you will have to split  in half. For
       example if you have n = 50 data points, then a 95% confidence interval will have  = 0.05
       and the t-statistic that you will be looking up in the table is   t             t 0.05           t 0.025, 49 .This
                                                                              , n 1            , 501
                                                                          2                 2
       value is not in the table since there is not a row for 49 degrees of freedom… but using a
       computer would tell us that this is about 2.0096).

       Finally, read example 17 on page 329, example 45 on page 334, and complete 8.1.1 on page
       340. The computation question on the exam will be similar to these sample problems.

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