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AP®︎/College Statistics
Unit 10: Lesson 1
Introduction to confidence intervalsInterpreting confidence levels and confidence intervals
AP.STATS:
UNC‑4.F (LO)
, UNC‑4.F.1 (EK)
, UNC‑4.F.2 (EK)
, UNC‑4.F.3 (EK)
, 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.
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.
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.
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In the try it for yourself exercise, what are the blue and red curves??(7 votes)
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)
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)
Great question! When you increase the sample size "n", the Margin of error decreases. This is because the formula for Margin of Error (in proportions) is the critical value times the standard error. The standard error is sqrt (phat)(1-phat)/n, where n is the sample size. So, as you increase n which is in the denominator, the standard error decreases, which means that the margin of error decreases. A link which I found helpful is https://www.dummies.com/education/math/statistics/how-sample-size-affects-the-margin-of-error/(2 votes)
Why don't we use a 100% confidence interval?(2 votes)
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)
- 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)
- recall how we calculate the standard deviation:
sqrt(p*(1-p)/n)
this is an estimation of the SD of the *population*
so the confidence level it construct doesn't work for the sample data(1 vote)
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)
What is this exactly ?(1 vote)- 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)
- What is the implication of not being able to have 100% confidence mean to performing an analysis?(1 vote)
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)
Is it true that an interval with a higher confidence level would have been a more preferable, and more informative?(1 vote)- 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)
- There is a blue and red curve on the bottom demonstration. What do they each represent?(1 vote)