confidence-intervals-of-polls

[confidence-intervals-of-polls]

UPDATE: I WAS ABLE TO TAKE IT AGAIN!!!

Just last week my Dad let me know when I was on the bus that they had scheduled me for a makeup test that same day. Thankfully I had enough time to take the Khan Academy course challenge again, and only stumbled on a couple MCQs! Somehow my last FRQ was super easy, did anyone else get the one about school schedules and power curves?

In any case, I'm confident on my test and will update in July on my score!

(Also I may be able to make this blog into an exhibition project next year so that's cool :D)

yo! how'd your ap stats exam go :0 i feel like it went pretty well for me, other than a couple mcq questions with stuff my teacher never talked about 😭 but otherwise good

Okay storytime. Important text is highlighted in orange.

So for context, I menstruate. I think you can see where this is going.

I'm taking the test, right? It's going well, all things considered. All of a sudden, I start getting cramps. BAD ONES.

If you're like me and you menstruate, you know that that shit hurts, and it can get very bad.

So I get some ibuprofen. IT DOESN'T RESPOND. And eventually it gets to the point where I have neither the physical or mental bandwidth to continue, and bailed after the first FRQ.

Now here's the thing: I COULD STILL PASS. Given what I've done I could still get a 3. But just to be safe we're filing an appeal so I might be able to get a retest, but the College Board gets the final say.

In the meantime SEND ME SOME POLLS SO I CAN PRACTICE MY INFERENCES. I NEED ALL OF IT FOR THE AP TEST

[confidence-intervals-of-polls]

yo! how'd your ap stats exam go :0 i feel like it went pretty well for me, other than a couple mcq questions with stuff my teacher never talked about 😭 but otherwise good

Okay storytime. Important text is highlighted in orange.

So for context, I menstruate. I think you can see where this is going.

I'm taking the test, right? It's going well, all things considered. All of a sudden, I start getting cramps. BAD ONES.

If you're like me and you menstruate, you know that that shit hurts, and it can get very bad.

So I get some ibuprofen. IT DOESN'T RESPOND. And eventually it gets to the point where I have neither the physical or mental bandwidth to continue, and bailed after the first FRQ.

Now here's the thing: I COULD STILL PASS. Given what I've done I could still get a 3. But just to be safe we're filing an appeal so I might be able to get a retest, but the College Board gets the final say.

In the meantime SEND ME SOME POLLS SO I CAN PRACTICE MY INFERENCES. I NEED ALL OF IT FOR THE AP TEST

[confidence-intervals-of-polls]

I'M LIVE

BACK FOR MORE PEEPS :D

Join me on Twitch as I figure out how to make the Fox Den a little cozier! Chatting encouraged, ideas welcome!

[confidence-intervals-of-polls]

AND WE ARE LIVE! Join me for Minecraft Survival on Twitch!

[confidence-intervals-of-polls]

AND WE ARE LIVE! Join me for Minecraft Survival on Twitch!

[confidence-intervals-of-polls]

Kinda the backbone of my blog so

(I can do other polls but these ones are easier)

[confidence-intervals-of-polls]

MARK YOUR CALENDARS

Guess who's back on Twitch? BIBS IS BACK ON TWITCH

On this coming Tuesday, April 15th, and the following Friday, April 18th, at 5PM Eastern Daylight Time (10PM Coordinated Universal Time) I will be streaming Minecraft on Twitch! Whether you're looking for something to have in the background, funny moments, or just a place to chat and have fun watching me fail as a Bedrock native playing Java, come join me on my Twitch once more!

[confidence-intervals-of-polls]

Hi there! Confidence Interval person here. Let's analyze this.

CHOOSE:

Our parameter, or p, that we're trying to estimate is the true proportion of all active [tumblr] users who say that they are frequently overstimulated.

For this, we're going to use a 1-sample z-interval for p, and our confidence level will be 99.9%.

CHECK:

As before, trying to make an online poll random is extremely difficult due to Voluntary Response Bias. As such, we will ignore the Random Condition, because if we took selection bias into account, nothing would go anywhere on here.

10% Condition: The average weekly active number of [tumblr] users is approx. 19.3 million. ~60,000 is most definitely less than 10% of the full population, and thus any dependence on responses is negligible in the sample.

Large Counts Condition: Again, I'm going to skip the calculations for this because the sample size and proportion is large enough to justify that going through the bother of working it out will be a waste of time.

CALCULATE:

The general formula for confidence intervals is point estimate ± margin of error, with the point estimate being our sample proportion and our margin of error being how much leeway we give the interval.

The specific formula for confidence intervals is: ˆp±z*sqrt((ˆp(1-ˆp))/n), where ˆp is our sample proportion, z* is our critical value (determined by our confidence level), and n is our sample size.

Now, we plug in our values and get:

0.758±3.291sqrt((0.758(1-0.758))/59421)

The rest is simply whittling it down to our two values that define the interval.

0.758±3.291sqrt((0.758(1-0.758))/59421)

0.758±3.291sqrt((0.758(0.242))/59421)

0.758±3.291sqrt((0.183436)/59421)

0.758±3.291sqrt(0.000003096436589)

0.758±3.291(0.00175966945)

0.758±0.00579107217

(0.75220892782, 0.76379107217)

CONCLUDE:

We are 99.9% confident that the interval between 0.752209 and 0.763791 (75.22% and 76.379%) captures the true proportion of all active [tumblr] users who say they are frequently overstimulated.

TL;DR:

Poll is accurate.

are you frequently overstimulated?

[confidence-intervals-of-polls]

as a fellow AP stats student i just wanted to say this is pretty cool!! i hope your stats teacher is proud

He said "Very nice!" in response to this :D

[confidence-intervals-of-polls]

AP Stats student here, it's more than that!

All polls on [tumblr] suffer from voluntary response bias. Online polls are generally like that, and it's why I ignore the random sample condition in my calculations. Plus, especially on polls that don't have any sort of "results" button, some people might pick a random one just so they can see the results, and that racks up quickly coming from lots of people.

In short, Getting reliable results on the internet is VERY hard to prove.

Do you believe the results of these polls suffer inevitably from selection bias?

[confidence-intervals-of-polls]

Hey there! Confidence Interval person here. Let's analyze this poll!

CHOOSE:

Our parameter, or p, that we're trying to estimate is the true proportion of all active [tumblr] users who say that they have cried over the death of a fictional character.

For this, we're going to use a 1-sample z-interval for p, and our confidence level will be 99.9%.

CHECK:

As before, trying to make an online poll random is extremely difficult due to voluntary response bias. As such, we will ignore the Random Condition, because if we took selection bias into account, nothing would go anywhere on here.

10% Condition: The average weekly active number of [tumblr] users is approx. 19.3 million. ~26,000 is most definitely less than 10% of the full population, and thus any dependence on responses is negligible in the sample.

Large Counts Condition: Our larger proportion is definitely over 10, but I hesitated to completely ignore this due to the landslide win, so I ran the calculations just to be sure.

n(0.937) = 24447.267 ≥ 10 ✅

n(1-0.937) = n(0.063) = 1643.733 ≥ 10 ✅

So we can assume the distribution is approximately normal.

CALCULATE:

The general formula for confidence intervals is point estimate ± margin of error, with the point estimate being our sample proportion and our margin of error being how much leeway we give the interval.

The specific formula for confidence intervals is: ˆp±z*sqrt((ˆp(1-ˆp))/n), where ˆp is our sample proportion, z* is our critical value (determined by our confidence level), and n is our sample size.

Now, we plug in our values and get: 0.937±3.291sqrt((0.937(1-0.937))/26091).

The rest is simply whittling it down to our two values that define the interval.

0.937±3.291sqrt((0.937(0.063))/26091)

0.937±3.291sqrt(0.059031/26091)

0.937±3.291sqrt(0.000002262504311)

0.937±3.291(0.00150416232)

0.937±0.00495019822

(0.93204980178, 0.94195019822)

CONCLUDE:

We are 99.9% confident that the interval between 0.932050 and 0.941950 (93.205% and 94.195%) captures the true proportion of all active [tumblr] users who have cried over the death over a fictional character.

EXTRA NOTES:

Finally I've done one where I don't immediately think I've made a mistake! These make much more sense than my other original values. Sorry for misleading some people!

TL;DR:

No significant evidence that the poll is inaccurate.

*This poll was submitted to us and we simply posted it so people could vote and discuss their opinions on the matter. If you’d like for us to ask the internet a question for you, feel free to drop the poll of your choice in our inbox and we’ll post them anonymously (for more info, please check our pinned post).

[confidence-intervals-of-polls]

HI AGAIN HERE TO CORRECT MY MISTAKE

CONTINUED CALCULATIONS:

0.336±3.291(sqrt(0.00005519643740))

0.336±3.291(0.00742943048)

0.336±0.02445025574

(0.31154974425, 0.36045025574)

REVISED CONCLUSION:

We are 99.9% confident that the interval between 0.311550 and 0.360450 (31.155% and 36.045%) captures the true proportion of active [tumblr] users who say that they would relive the last ten years, if given the chance.

EXTRA NOTES:

This makes so much more sense. Glad I caught that mistake early on in this blog so I didn't have to go back and change a ton of stuff.

TL;DR:

Fixed a mistake. Poll is still accurate.

[confidence-intervals-of-polls]

*BREAKS DOWN THE DOOR* I FORGOT A STEP I'M SO SORRY STATS NERDS

REVISING MY CALCULATIONS. THE STANDARD ERROR NEEDS TO BE SQUARE ROOTED.

0.751±3.291 • sqrt(0.000001808640900)

0.751±3.291(0.00134485720)

0.751±0.00442592505

(0.74657407495, 0.75542592505)

REVISED CONCLUSION:

We are 99.9% confident that the interval between 0.746574 and 0.755426 (74.6574% and 75.5426%) captures the true proportion of active [tumblr] users who think they could take a vampire.

EXTRA NOTES:

That makes a LOT more sense. Although, I'm still surprised of how small of a dent it made.

TL;DR:

Made a mistake. Poll is still pretty accurate.

do you think you could take a vampire?

[confidence-intervals-of-polls]

Hi there! I'm here to evaluate this poll and see if, based on these results, we can find something that captures the true proportion that this poll couldn't quite get to in time.

CHOOSE:

Our parameter, or p, that we're trying to estimate is the true proportion of all active [tumblr] users who say that they would relive the last ten years, if given the chance.

For this, we're going to use a 1-sample z-interval for p, and our confidence level will be 99.9%.

CHECK:

As before, trying to make an online poll random is extremely difficult. As such, we will ignore the Random Condition, because if we took selection bias into account, nothing would go anywhere on here.

10% Condition: The average monthly active number of [tumblr] users is approx. 135 million. ~4,000 is most definitely less than 10% of the full population, and thus any dependence on responses is negligible in the sample.

Large Counts Condition: Again, I'm going to skip the calculations for this because the sample size and proportion is large enough to justify that going through the bother of working it out will be a waste of time.

CALCULATE:

The general formula for confidence intervals is point estimate ± margin of error, with the point estimate being our sample proportion and our margin of error being how much leeway we give the interval.

The specific formula for confidence intervals is: ˆp±z*((ˆp(1-ˆp))/n), where ˆp is our sample proportion, z* is our critical value (determined by our confidence level), and n is our sample size.

Now, we plug in our values and get: 0.336±3.291((0.336(1-0.336))/4042).

Now, we just simplify the expression down to our two numbers that represent the interval.

0.336±3.291((0.336(0.664))/4042)

0.336±3.291(0.223104/4042)

0.336±3.291(0.00005519643740)

0.336±0.00018165147

(0.33582834852, 0.33618165147)

CONCLUDE:

We are 99.9% confident that the interval between 0.335828 and 0.336182 (33.5828% and 33.6182%) captures the true proportion of active [tumblr] users who say that they would relive the last ten years, if given the chance.

EXTRA NOTES:

...I really need to find polls with smaller sample sizes. This is fun, but if it's making such a tiny dent it's kinda underwhelming, huh?

TL;DR:

The poll is accurate. No convincing evidence says otherwise.

[confidence-intervals-of-polls]

You're welcome! I just found confidence intervals cool, so I made a blog that just does confidence intervals for the fun of it.

do you think you could take a vampire?

[confidence-intervals-of-polls]

The problem with Tumblr polls is, as I said, they only ever reach a subset of Tumblr via searches, reblogs, etc. I went into this knowing said bias was going to be present no matter what.

Selection bias for online polls is extremely difficult to avoid, and I'm honestly unsure how I'd account for it or get around it. I'm just a student in AP Statistics, so this is a generally new concept for me and I just made this for fun and practice.

If you have any methods you'd like to bring up, please do share them with me! I'd love to understand more about how to avoid this sort of bias with polls.

do you think you could take a vampire?

[confidence-intervals-of-polls]

Hi there! Let's see if we can break this down and find something that can capture the true proportion of ALL active [tumblr] users who think they could take a vampire.

CHOOSE:

The Parameter we're looking to estimate for is p, which is the true proportion of all active [tumblr] users who think they could take a vampire.

How are we going to do this? Using a one-sample z-interval for proportions.

Because our sample size is SO large, I'm going to go with a 99.9% confidence level for our interval, just to make it actually do something (the higher your confidence level, the wider the interval).

CHECK:

To be entirely fair, there's no way to make a poll sample random people, so we are going to ignore the Random rule for now.

10% Condition: The average monthly active number of [tumblr] users is approx. 135 million. ~103,000 is most definitely less than 10% of the full population, and thus any dependence on responses is negligible in the sample.

Large Counts Condition: I'm going to skip the calculations for this simply because the sample size and proportion is large enough to justify that it's going to pass.

CALCULATE:

The general formula for confidence intervals is point estimate ± margin of error, with the point estimate being our sample proportion and our margin of error being how much leeway we give the interval.

The specific formula for confidence intervals is: ˆp±z*((ˆp(1-ˆp))/n), where ˆp is our sample proportion, z* is our critical value (determined by our confidence level), and n is our sample size.

Now, we plug in our values and get: 0.751±3.291((0.751(1-0.751))/103392).

Now for the busywork.

0.751±3.291((0.751(0.249))/103392)

0.751±3.291(0.186999/103392)

0.751±3.291(0.000001808640900)

0.751±0.000005952237204

(0.75099404776, 0.75100595223)

CONCLUDE:

We are 99.9% confident that the interval between 0.75099404776 and 0.75100595223 (75.0994% and 75.1006%) captures the true proportion of active [tumblr] users who think they could take a vampire.

EXTRA NOTES:

Okay, I severely overestimated how the sample size would sway this. The confidence interval is negligible, even with a 99.9% confidence level. But I do hope this taught you a thing or two about statistics.

TL;DR:

The poll is accurate. No significant reason to think it's not.

do you think you could take a vampire?