# hypothesis testing: proportions with 3 samples

#### hurzz

##### New Member
There are 3 hotels. In each, customers were asked if they would return for a second stay:

How could I test if there is a significant difference in the proportion of customers that would return for each hotel?

#### Karabiner

##### TS Contributor
What would you suggest?

With kind regards

Karabiner

#### hurzz

##### New Member
What would you suggest?

With kind regards

Karabiner
Maybe I could get the samples two by two and perform a hypothesis test to check if the difference of the proportions in between these two are:
Ho: difference in proportions=0
Ha: difference in proportions !=0;
So I would end up with 3 hypothesis tests (Golden Palm x Palm Royale; Golden Palm x Palm Princess; Palm Royale x Palm Princess)

but I'm wondering if there is a more straightfoward way to do that!

#### joeb33050

##### Member
I would add to the table; it looks like no %NO is even close to another.
Then check the Binomial, and find that the probability of any two %NO is ~Zero.

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#### joeb33050

##### Member
"How could I test if there is a significant difference in the proportion of customers that would return for each hotel?"
All you need do is calculate NO/REPLIES, 41%, 14% and 26%, there are obviously significant differences between each pair.
Its easy.
joe b.

#### Dason

"How could I test if there is a significant difference in the proportion of customers that would return for each hotel?"
All you need do is calculate NO/REPLIES, 41%, 14% and 26%, there are obviously significant differences between each pair.
Its easy.
joe b.
You are wrong. If that was the case there would be no need for statistics.

#### joeb33050

##### Member
You are wrong. If that was the case there would be no need for statistics.
No, Dason, you are wrong. I learned many years ago, and taught my students, that there are two times when statistics is inappropriate.
This case is one where we know everything we need to know by looking at the data.. We can dazzle the uninformed, bit bring nothing to the table.
The second is where we lack the data to make a reasonable statement-we need to stop, quit, or get more data.
Statistics is like a hammer, just because you have one doesn't mean you have to hit something.
joe b.

#### spunky

##### Doesn't actually exist
No, Dason, you are wrong. I learned many years ago, and taught my students, that there are two times when statistics is inappropriate.
This case is one where we know everything we need to know by looking at the data.. We can dazzle the uninformed, bit bring nothing to the table.
The second is where we lack the data to make a reasonable statement-we need to stop, quit, or get more data.
Statistics is like a hammer, just because you have one doesn't mean you have to hit something.
joe b.
And how exactly do you account for sampling error if you only offer descriptives?

#### Dason

No, Dason, you are wrong. I learned many years ago, and taught my students, that there are two times when statistics is inappropriate.
This case is one where we know everything we need to know by looking at the data.. We can dazzle the uninformed, bit bring nothing to the table.
The second is where we lack the data to make a reasonable statement-we need to stop, quit, or get more data.
Statistics is like a hammer, just because you have one doesn't mean you have to hit something.
joe b.
So your solution is to not try anything? If your advice is just to merely look at the sample estimates and then call it a day then I'm not sure why you feel like you should be dispensing advice. Accounting for variability is the reason statistics as a field exists. If you have a specific reason in this case why you think doing something like a chi-square test is inappropriate then I'd be happy to discuss your objections. But if all you have is "yeah it looks like there are differences so obviously there are" then you're just plain wrong. I could come up with lots of examples where just looking at the data might trick some people into thinking there truly are differences even though there aren't.

#### hlsmith

##### Less is more. Stay pure. Stay poor.
Beyond sampling error, I would ask whether these are random samples and whether subjects are exchangeable between groups?

If the generic question was whether there are differences between the groups in responses, I would define my level of significance, then conduct the three rate difference calculations and then adjust for false discovery in the confidence intervals and plot. Or if using a Bayesian approach repeat the process and plot the three posteriors distributions for rate differences. Three comparisons are likely the most informative result compared to binomial test with just p-values or looking at the standardized residuals from chi-sq.

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#### Miner

##### TS Contributor
"How could I test if there is a significant difference in the proportion of customers that would return for each hotel?"
All you need do is calculate NO/REPLIES, 41%, 14% and 26%, there are obviously significant differences between each pair.
Its easy.
joe b.
It is not always as clear cut as you propose. While this specific case does show distinct differences, note the wide confidence intervals around each hotel's proportions. Slightly smaller sample sizes and/or smaller effect sizes and the differences would not be statistically different.

#### joeb33050

##### Member
It is not always as clear cut as you propose.

Did I propose that it is always as clear cut as the example? Is this the third grade?

While this specific case does show distinct differences, note the wide confidence intervals around each hotel's proportions. Slightly smaller sample sizes and/or smaller effect sizes and the differences would not be statistically different.

So your solution is to not try anything? If your advice is just to merely look at the sample estimates and then call it a day then I'm not sure why you feel like you should be dispensing advice. Accounting for variability is the reason statistics as a field exists. If you have a specific reason in this case why you think doing something like a chi-square test is inappropriate then I'd be happy to discuss your objections. But if all you have is "yeah it looks like there are differences so obviously there are" then you're just plain wrong. I could come up with lots of examples where just looking at the data might trick some people into thinking there truly are differences even though there aren't.
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#### joeb33050

##### Member
Beyond sampling error, I would ask whether these are random samples and whether subjects are exchangeable between groups?

If the generic question was whether there are differences between the groups in responses, I would define my level of significance, then conduct the three rate difference calculations and then adjust for false discovery in the confidence intervals and plot. Or if using a Bayesian approach repeat the process and plot the three posteriors distributions for rate differences. Three comparisons are likely the most informative result compared to binomial test with just p-values or looking at the standardized residuals from chi-sq.
I conclude that the means differ, all 3 and all pairs. I'm working without a net.
Why don't you show us that I'm incorrect. Anybody?

#### hlsmith

##### Less is more. Stay pure. Stay poor.
Incorrect in regards to what?