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# Data inferences — Harder example

Watch Sal work through a harder Data inferences problem.

We make a confidence interval by starting with a sample result and adding and subtracting the margin of error. Consequently, the sample result is the exact middle of the interval. If we were to build a new interval based on the same data but with a reduced confidence level, the new interval would have the same center but a smaller margin of error. Here's a diagram illustrating this problem another way.

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## Want to join the conversation?

• Is there any way to solve this via calculation, rather than 'These sound shady, and I like the sound of this choice, so we'll go with this' ?? •   Yes, a confidence interval is defined by:
sample mean ± margin of error

It said the the 95% confidence interval is from 22.76 to 59.24
Using this we can find the sample mean which would just be the mean of these two numbers: (59.24 + 22.76) / 2 = 41

Moving across confidence intervals will only affect the margin of error but won't affect the sample mean so the 90% confidence interval would still have to have a sample mean of 41 and take into consideration that the range would decrease because were decreasing the confidence interval. Now we just test the answers given:

17.10 to 64.90: (17.10 + 64.90) / 2 = 41
This solution does have the same sample mean but is has a much greater range than the original interval so it would be more precise rather than less precise so this can't be the answer.

20.48 to 53.32: (53.32 + 20.48) / 2 = 36.9
This isn't 41 so we can rule this answer out.

21.56 to 56.12: (21.56 + 56.12) / 2 = 38.84
This isn't 41 so we can rule this answer out

25.65 to 56.35: (25.65 + 56.35) / 2 = 41
This answer does have a sample mean and has a lesser range than the 95% confidence interval so it is the correct answer.
• Hi, I thought that the correct choice would be 3 due to the following reasons:
Basically, the range (difference between max possible value and min possible value) which the median could be in with a 95% confidence level is
59.24 - 27.76 = 36.48
Therefore, the range which the median could be in with a 90% confidence level should be
90/95 * 36.48 = 34.56

Now, if we calculate the range for all the choices:
Choice 1 : 64.90 - 17.10 = 47.8
Choice 2: 53.32 - 20.48 = 32.84
Choice 3: 56.12 - 21.56 = 34.56
Choice 4: 56.35 - 25.65 = 30.7

Therefore, we see that the range in choice 3 is exactly the same as the range I calculated previously for 90% confidence level, therefore the correct answer is choice 3. Can anyone explain what I am getting wrong here? • You are pretty accurate about the difference of the two numbers, but an important factor you didn't consider is where the numbers are on the number line. Example: 41.56 and 76.12 also have the same difference, i.e. 34.56 but we know that our confidence level should be much less than 90% in this case.
• I think that if your confidence level is lower, then you will choose a broader range, you wouldn't be sure of the exact value and would make assumptions. If your confidence level is 100%, you will know the exact median, but if it is 0 %, then u may say that any observation could be the median. As your confidence level will increase, you would narrow down to get closer to the median until u find the exact value. At this point of time, you will obviously have 100 percent confidence. But, if u follow the video's logic, when u will have 100 % confidence (which means u know the answer), instead u will consider the whole data set as the range for the median! What do you think? • could you please explain how do we actually calculate the confidence level for a given data estimate? • So...what makes choices 2 and 3 incorrect? Sal just said, "I like this choice" & then picked choice 4.... • The main issue with 2 and 3 is that they offer values that are outside the range given in the question, which brings up problems about the distribution of numbers which we can't answer with the info given.

For example, it's possible there is a big gap between the lowest score in the 95% confidence range (\$22.76) and the next lowest data point. It could be \$15. If that were the case, expanding the range to \$20 would have no effect on the median. If there are a ton of data points between 22.76 and 20 compared to the other end of the range, it could shift the median. But again, we have no way of knowing, so we can't assume.

Choice 4 offers two numbers that are inside the range given for 95% confidence. That way, we haven't added any new values and the potential statistical issues, we're just narrowing the range of numbers we know contain the answer (95% sure). Thus, it's the only answer that MUST have a lower confidence than the original set.
• what in the world is a confidence level • Can we do this type of question by following way?
solution:
Let x be the median hourly wage at 100% confidence level.
By question,
95% of x = 22.76 or 59.24 ( you can do any one part of it)
so x = approx. 23.96 or 62.36
Now,
90 % of x ( approx 23.96 or 62.36) = approx. 21.56 or 56.12 • Alright, I'm a bit conflicted now. Because I tried doing the question before watching, for practice, and I used a calculator to be specific (though I still don't know whether we're allowed to use calculators in Sal's examples.)
My working was simply: 90/95 * \$22.76, and 90/95 * \$59.24.
My answer was the third one, not the fourth. • Bro you can't just eyeball it and tell us the answer, teach us the equation and how to apply it because we obviously can't eyeball it on the real SAT like you did here. • Actually, you can just eyeball it. Check out the link; at its most basic level, we need to find the answer choice that-
A) Has the same sample median and B) has a smaller range

A) needs to be true because it is based on the same sample, this means that the range is not going to be shifting left and right as a whole, it's just going to be getting "wider" or "smaller".
B) needs to be true because there is less of a chance that the range contains the true median hourly wage. ("Lower confidence level") The only way there can be less of a chance is for the range to be smaller. 