3. Confidence Intervals

Learning objectives (and summaries)

Estimate population parameters with a confidence interval using simulation.
  • Understand that a sampling distribution is a collection of many possible statistics
    • For example, it is a bunch of means of different samples thrown in one histogram together.  Not all sample means are the same.
    • The main reason we care about this is to see how spread apart the different possible means are from each other.  This helps us make confidence intervals.
  • Understand the purpose of a confidence interval
    • To give a clearer prediction of the parameter (the mean or proportion of the population).  A single number says little about the precision of the guess, but 95% confidence that the mean is between 12.2" and 15.8" is much more informative.
  • Use bootstrap simulation to predict a confidence interval for means
    • Use the StatKey simulator with a set of sample numbers and thousands of resamples to create a simulated sampling distribution.  Use this to identify the ___% confidence interval of the population mean.
  • Convert between interval and ± forms of confidence intervals
    • For example, 12.2% to 15.8% can be rewritten as 14% ± 1.8%.  The second form is often nicer to read since you can see the middle estimate (14%) and the margin of error (1.8%).
  • Communicate a confidence interval of a mean in a sentence
    • CI of means: I am _(__)_% confident that the average _(_____)_ of _(_____)_ is between _(___)_ and _(___)_.
    • For example: I am 95% confident that the average weight of dogs in Minnesota is between 17lbs and 26lbs.
Instruction

      Last two are review videos from previous section (Confidence Intervals of Proportions).  Watch if you forgot how to convert plus-minus form (first one) or struggle to explain a CI of a mean in a sentence (second video).

      For more material from the New Zealand lady (Dr. Nic), check out this page.

      Essay Questions
        • There are no essay questions for this topic
        Vocabulary
            bootstrapping- Used to find hypothesis tests and confidence intervals; it takes your set of data and uses it as the population and uses it to do a bunch of tests so that you get more results

            Practice
                1. A group of students decided to track the average number of touches by both teams in each point in volleyball games in the HVL this year.  They did an SRS of which games to watch and then used systematic sampling to decide which plays to record data about.  The list of values, in number of touches: 4, 3, 17, 12, 1, 19, 7, 26, 7, 3, 9, 15, 28, 11, 7, 3, 13, 8.  [Copyto StatKey from here.]
                • a. Are the sampling conditions met so that we can actually perform a meaningful confidence interval?
                • b. What is the population that we're studying?
                • c. What is the variable that we're studying?  Is it quantitative or categorical?  Is it summarized with a mean or a proportion?
                • d. Copy the data into StatKey and perform the appropriate type of confidence interval at 95% confidence using at least 1000 bootstrap samples.  What is your interval?
                • e. Explain all that you just found in a sentence.
                • f. Rewrite your interval in +/- form.
                • g. Explain your findings in a sentence using the +/- form of the interval.
                • h. Did your interval capture the true mean/proportion of your population?  How would you check?
                2. A group of players decide to track the average number of goals in each soccer game in the state this year. They did an SRS of which games to watch and then used systematic sampling to decide which teams to record data about. The list of values, in number of goals: 2, 4, 3, 1, 6, 3, 9, 2, 3, 5, 2, 5, 1, 2, 1, 4. (Use StatKey)
                • a. Are the sampling conditions met so that we can actually perform a meaningful confidence interval?
                • b. What is the population that we're studying?
                • c. What is the variable that we're studying?  Is it quantitative or categorical?  Is it summarized with a mean or a proportion?
                • d. Copy the data into StatKey and perform the appropriate type of confidence interval at 95% confidence using at least 1000 bootstrap samples.  What is your interval?
                • e. Explain all that you just found in a sentence.
                • f. Rewrite your interval in +/- form.
                • g. Did your interval capture the true mean/proportion of your population?  How would you check?

                Practice solutions
                    1. Volleyball touches
                    • a. Yes -- SRS and systematic sampling are good, random techniques
                    • b. Volleyball games in the HVL
                    • c. Number of touches per play.  This is a quantitative variable, so we're looking of the mean.
                    • d. 7.44 to 14.44
                    • e. I'm 95% confident that the average number of touches per play at HVL volleyball games is between 7.44 and 14.44 touches.
                    • f. 10.94 ± 3.5 (work below)
                      • Middle: (14.444 + 7.44 )/2 = 10.94
                      • Margin of error: (14.444 - 7.44 )/2 = 3.5
                    • g. I'm 95% confident that the average number of touches per play at HVL volleyball games is 10.94 ± 3.5 touches.
                    • h. Who knows?  The only way to check a confidence interval is to do a census -- track EVERY play of every conference game.  If we had time to do that we wouldn't bother with confidence intervals, so you just have to trust that stats works at there is a 95% chance that we got it right.

                    2. Soccer Goals

                    a. Yes, SRS and systematic sampling are good, random techniques

                    b. Soccer goals in the State

                    c. Average number of goals per game. This is a quantitative variable, so were looking for the mean.

                    d. 2.375 - 4.375

                    e. Im 95% confident that the average number of soccer goals per game is between 2.375 and 4.375.

                    f. 3.55+- 1.15

                    g. There is no way to tell, the only way to check a confidence interval is to do a census.



                    Notes