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Interpreting confidence levels and confidence intervals

AP.STATS:
UNC‑4.F (LO)
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UNC‑4.F.1 (EK)
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UNC‑4.F.2 (EK)
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UNC‑4.F.3 (EK)
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UNC‑4.F.4 (EK)
When we create a confidence interval, it's important to be able to interpret the meaning of the confidence level we used and the interval that was obtained.
The confidence level refers to the long-term success rate of the method, that is, how often this type of interval will capture the parameter of interest.
A specific confidence interval gives a range of plausible values for the parameter of interest.
Let's look at a few examples that demonstrate how to interpret confidence levels and confidence intervals.

Example 1: Interpreting a confidence level

A political pollster plans to ask a random sample of 500 voters whether or not they support the incumbent candidate. The pollster will take the results of the sample and construct a 90, percent confidence interval for the true proportion of all voters who support the candidate.
Which of the following is a correct interpretation of the 90, percent confidence level?
Choose all answers that apply:
Choose all answers that apply:

Example 2: Interpreting a confidence interval

A baseball coach was curious about the true mean speed of fastball pitches in his league. The coach recorded the speed in kilometers per hour of each fastball in a random sample of 100 pitches and constructed a 95, percent confidence interval for the mean speed. The resulting interval was left parenthesis, 110, comma, 120, right parenthesis.
Which of the following is a correct interpretation of the interval left parenthesis, 110, comma, 120, right parenthesis?
Choose all answers that apply:
Choose all answers that apply:

Example 3: Effect of changing confidence level

Suppose that the coach from the previous example decides they want to be more confident. The coach uses the same sample data as before, but recalculates the confidence interval using a 99, percent confidence level.
How will increasing the confidence level from 95, percent to 99, percent affect the confidence interval?
Choose 1 answer:
Choose 1 answer:

Want to join the conversation?

  • leafers tree style avatar for user Jared Bailey
    In the try it for yourself exercise, what are the blue and red curves??
    (7 votes)
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    • leaf green style avatar for user B1-66ER
      The red line is the sample distribution, the blue line is the population distribution. The reason why the blue line changing it's shape while adjusting sample size is the scale of the whole chart is changing, which means the blue line actually isn't changing at all, just the zooming out to in order to show the full area of red line, I think.
      (4 votes)
  • blobby green style avatar for user owen-k
    Drawing more samples causes the interval to narrow, lowering the confidence level also causes the confidence interval to narrow. But I think a 99% confidence level means you are more certain that your population parameter would fall into that interval, right? So why is a narrow interval in terms of a higher sample size 'good' but a narrow interval in terms of a smaller confidence level 'bad'?
    (4 votes)
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  • hopper cool style avatar for user Mark Ionkin
    Why don't we use a 100% confidence interval?
    (2 votes)
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    • mr pants teal style avatar for user Soo Kyung Ahn
      The normal distribution is defined from negative infinity to positive infinity and the corresponding 100% confidence interval would be from negative infinity to positive infinity as well. It doesn't provide useful information, and thus it is not used.
      (4 votes)
  • blobby green style avatar for user grace9570
    In Example 2, shouldn't B also be correct? To construct the 95% confidence interval, we add/subtract 2 standard deviations from the mean. Given the distribution of the sample is approximately normal, this interval would also contain about 95% of the sample pitches.
    (3 votes)
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  • aqualine ultimate style avatar for user jayceelagula
    So the best way to estimate the population mean is to set a higher confidence level AND increase the sample size if you want a good balance of how confident you are that the interval captures the true mean and how accurate that interval might be (i.e. low standard error) - is that correct?
    (1 vote)
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  • winston default style avatar for user Abbas Al-bayati
    What is this exactly ?
    (1 vote)
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    • blobby green style avatar for user Sanjana Khedekar
      This page and simulation shows what confidence interval tells us about estimation of population mean, when population mean is not given.
      For example, if we keep 95% confidence interval and take confidence levels of many sample sets, each of these confidence level will show us if it truly captured the estimated mean.
      To ensure the above, typically 95% confidence level is desired with large sample size.
      (1 vote)
  • aqualine ultimate style avatar for user sassyshourya13
    What is the implication of not being able to have 100% confidence mean to performing an analysis?
    (1 vote)
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  • aqualine sapling style avatar for user ajvkrish
    From the sample, we can say from the mean and standard deviation that 95% of the sample data would fall between two points. How can we generalize that to say that the population mean will fall between the two points 95% of the time?
    (1 vote)
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  • aqualine seed style avatar for user I will eat you
    Is it true that an interval with a higher confidence level would have been a more preferable, and more informative?
    (1 vote)
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    • blobby green style avatar for user zjleon2010
      It depends on the context of problem, but in my opinion, it not.
      Sometimes it may require you to give a precise estimation, i.e, a higher confidence level
      In other circumstances, a lower confidence level with narrower interval is acceptable, because it gives us a executable plan and how like it going to work.
      Say you need to develop a new product next year, and you need a finance plan for it.
      Like always, you don't have infinite amount of money to support it, so given that much amount of money you have, what's your expectation on the successes should be your prior concern
      (1 vote)
  • blobby green style avatar for user earl kraft
    There is a blue and red curve on the bottom demonstration. What do they each represent?
    (1 vote)
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