Confidence Intervals
As you noticed in the previous section, we can produce many trials to find means for the mileage of our cars. The next step is to determine how to use these means to determine the mean for the mileage of American cars. Since we do not know the mileage for all cars, we cannot determine exactly what the true mean is, based on our random sample. So the best we can do is determine an interval with a certain degree of confidence (or accuracy) for what this mean can be.
Progress through the questions below and use the worksheet "Day 2" to help you learn how to develop these confidence intervals.
Make sure the "Sampling Rate" in cell C2 is set to 15!
In this worksheet we will be able to generate many trials quickly. You can either push the "Run One, Ten or Hundred Trials" button to run that many trials. (Note: One hundred trials take about a minute to run, so do not press it unless you're willing to wait!).
- Generate ten trials. What is the range of the trials and how much "variability" is there in the trials?
- Generate a total of 20 trials. Compare it to the previous trial in terms of the range and the variability.
- Look at the frequency table in columns G and H. If a trial falls within plus or minus 0.5 of the number in column G, it is counted in column H.
- Generate 50 trials.
- How often do the trials fall within the range of 23.5 to 24.5?
- How about in the range 19.5 to 20.5?
Look at the histogram plot below the frequency table. This is simply a plot of the frequency table. Based on the plot,
- How many times did the trials fall within the range 24.5 to 25.5?
- How about in the range 29.5 to 30.5?
- What is the shape of the plot. Does it resemble a "bell" shape?
- Plots that resemble this shape are called "normal". Can you think of other types of data sets that would fit this shape?
- Now, generate 100 trials.
- Does the histogram still resemble a "normal" plot?
- What percent of the trials fall below 21?
- Based on your percentage, how well does it reflect the "true" population mean of the mileage of the cars?
- What percentage would you think be too low and too high for an accurate estimate of the car's mileage?
- To develop a confidence interval for the cars' mileage, we need to determine what percentage we believe is significant. Often, the choice is from 5-95% (90% of all trials) of the trials. Other times, it can be 2.5-97.5% or 0.5 -99.5% depending on how accurate we want our interval to be.
In the spreadsheet, click the "Find 95th and 5th percentiles" button. This will find the trials that are the 5th and 95th percent of all of the trials. What is the confidence interval for the mileage of American cars?
- Now, click the "Erase Trials" button. Generate 100 trials again and find the confidence interval for these trials. Compare your answer to the one you found previously.
- Erase the trials and generate 100 more trials. Find the confidence interval and compare it to the ones you found before.
- How stable are the intervals that you found? In other words, are you getting similar intervals or do they vary greatly?
- If someone told you that the true population mean of the mileage of American cars is 28, could you use the intervals you found to support or reject this statement.
Continue to next section.
Return to Introduction.