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### Course: Digital SAT Math > Unit 3

Lesson 9: Data inferences: foundations# 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.

.## 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' ??(160 votes)
- 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.(288 votes)

- 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?(56 votes)- 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.(17 votes)

- 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?(50 votes)
- I have to agree with you. I also understand Harman Arora's point, however, there was nothing in the question which specified the difference. The question is bad, plain and simple.(15 votes)

- Ok, So in the previous lessons about Margin of Error, We learnt about the impact of estimates on margin of errors and vice versa. For instance, how a larger estimation could result in a lower Margin of Error. Same is the rule applied here. Since, the confidence level is decreasing from 95% to 90%, this implies that the margin of error is increasing from 5% to 10% ! Therefore the interval/estimation/range tends to get smaller as in this case should be in between $22.76 & $59.24. Hope this helps.(28 votes)
- That's awesome, FATEH! Thanks so much! That really explains it super clear!(4 votes)

- could you please explain how do we actually calculate the confidence level for a given data estimate?(17 votes)
- Hi Mishra-

See if this does the trick. https://www.khanacademy.org/math/probability/statistics-inferential/confidence-intervals. I myself have not listened to these videos, so I'm not absolutely sure that this is the same concept as talked about in the SAT video. It does seem to fit the bill.(10 votes)

- honestly I didn't get this at all(13 votes)
- I didn't either but a lot of people in the comments have really great explanations.(8 votes)

- So...what makes choices 2 and 3 incorrect? Sal just said, "I like this choice" & then picked choice 4....(11 votes)
- 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.(7 votes)

- Honestly I do not agree, no offense, from the previous knowledge we can infer that decrease in confidence which is increase in uncertainty should lead to an increase in range causing it to be more diverse. Who can understand where I am going?(8 votes)
- I also thought the same, thus I would have chosen A as well; however, I now understand that lowering the confidence level meant to increase your margin for error, and to increase your margin for error you must pick the scenario with the smallest interval. You do this because, when you
**decrease the range of possible values**, you**increase your margin for error**, which is also the same as decreasing your confidence level.(12 votes)

- Im confused. If the confidence level is lower, shouldnt the range be bigger? When you are less confident about where the median is, the range should be bigger so its more likely that the median is in that range.(13 votes)
- Think of it the other way around.

How do you become less confident? If the range is narrower.(2 votes)

- this doesn't make sense is there some like more mathematical way to do this?(7 votes)
- You could take the
*ranges*of each of the options.

a) Range:*47.8*

b) Range:*32.84*

c) Range:*34.56*

d) Range:*30.7*

As mentioned that the*range*for**90% confidence level**should be a*subset*of the given*range*, meaning that it should be*smaller*than the given range. This can be seen as*part*(**d**) has the*smallest range*.

If we were asked to calculate a*range*but with a**higher confidence level**, lets say**98**%, then part (**a**) would have been the answer as it would have had the*greatest range*.

(not sure if this would work all the time, just something i thought of)(11 votes)

## Video transcript

- [Instructor] A researcher
collecting information about 1,000 randomly
selected physical therapists concluded that the median hourly wage for physical therapists
in the United States at the time of the study was between $22.76 and $59.24 with a 95% confidence level. Which of the following could represent the median hourly wage,
based on the same sample, same sample, for physical
therapists in the United States with a 90% confidence level? So let's just think about
what confidence level means. That means, remember,
the median is gonna be some number, it might be the
actual the median hourly wage for physical therapists. It might be $30 an hour, $25 an hour. They're trying to estimate it by doing this random selection and then they're providing a range and they're saying, hey,
there's some confidence level that this range captures
that true median hourly wage. So when they say there's
a 95% confidence level, they're saying that
there's a 95% probability that the true median is
between these two numbers. Now if we're talking about
a 90% confidence level, if we're talking about
a 90% confidence level that means we are less
confident that the true median is between these two numbers. In order to be less confident, you would wanna have
even a narrower range. You would want a range that is a subset of this range right over here. Let me make this clear. So the range is $22.76 all the way to $59.24. So they say it's a 95% confidence level. That means, I'm gonna actually
draw a number line here, so let's say these are just
points on the number line, these are points on the number line. So there's 22.76, this is 59.24, they're 95% confident that the true median that the true median is going to be that it's going to be
between these two values. So they're 95% confident
that the true median is there that the true median is there or that the true median is there but there's still a 5% chance that maybe the true media could be here. It could be below this
range or above this range. Now if we're talking about
a lower confidence level, 90% confidence level, that means that the reign, this range should be narrowed. If you're gonna be less confident, that means you wanna have, or the only way you're
gonna be less confident is if you have a narrower range. If you had a broader range, if it went from say here to here, you'd be even more confident. This type of a range you might
have a 97% confidence level. So at 90% less confident, you're looking at something that might
look something like that might look something like that. Now let's look at the choices. So this is $17 to 64.90, so this is actually more like this one. You're starting lower
and you're ending higher. So you should be even more if you're 95% confident
that this range capture you should be even more confident that this range captured it. So this would actually maybe be a 97 or 98 who knows confidence level. Not a 90% confidence level
so we can rule that out. 20.48 to 53.32. So this one's interesting because it starts lower but
then it ends lower too. So I don't know what you could actually what you could say it's based on the same I don't know, this one's
a little bit it depends kind of what the distribution
that you selected was, they didn't tell us a lot about that. This is a little bit, this
one feels a little bit shady. 21.56 to 56.12, this is also similar it starts a little bit lower and then it ends a little bit lower. So it's kind of shifted the range and so this is also, you don't know for sure
that you're going to have, I mean you're probably going to have a well, they don't tell us a
lot about the distribution so I'm not gonna make too
many statements there. Now this last one goes
from 25.65 to 56.35. So that's going to be something like that's going to be, actually,
like from here to here. This is going to be a narrower range. So you would be less confident. So this could be something
that represents a 90%, a 90% confidence level. So actually I would go with this one because this is kind of a purely this is a subset of the previous range so I like this choice right over here.