If the entire population of 2 sample groups is more than 30 but both sample groups are less than 30, do I use T-test or Z-test?

[Mackattack]

Hello. I'm comparing the test scores of boys and girls to see if there's a significant difference. I'm confused about whether I should use T-test or Z-test. There are 33 students—18 are boys and 15 are girls. I know that Z-test is used when n>30 but does that only apply if either boys or girls are more than 30, or if boys and girls combined are more than 30? Either way, both tests say to accept null hypothesis, and the critical values are not that far from each other. Additionally, can I still use T-test if the number of both girls and boys is 35? Thank you.

 

[Dr.Peterson]

Hello. I'm comparing the test scores of boys and girls to see if there's a significant difference. I'm confused about whether I should use T-test or Z-test. There are 33 students—18 are boys and 15 are girls. I know that Z-test is used when n>30 but does that only apply if either boys or girls are more than 30, or if boys and girls combined are more than 30? Either way, both tests say to accept null hypothesis, and the critical values are not that far from each other. Additionally, can I still use T-test if the number of both girls and boys is 35? Thank you.
View attachment 34584

Wherever you found the rule that n>30 for the Z test, check how they defined n! That's how you find out how to use the rule.

Equations and inequalities involving variables that are not defined are useless. Everything depends on their meaning.

I searched for a source and found none that both said n>30 and defined n. What they more often say is that in real life you use t, because z assumes you know the population standard deviations, which you don't here. If your source doesn't define n, you might stop to think about where else you have see the same inequality, and why it would be used here. That may suggest an answer to your question.

Or, you might consider the fact that the t test can always be used; the z test is making an assumption that is only justified under certain conditions (and perhaps not really even then). Do they give you any restrictions on when you can use t?

 

[Mackattack]

Wherever you found the rule that n>30 for the Z test, check how they defined n! That's how you find out how to use the rule.

Equations and inequalities involving variables that are not defined are useless. Everything depends on their meaning.

I searched for a source and found none that both said n>30 and defined n. What they more often say is that in real life you use t, because z assumes you know the population standard deviations, which you don't here. If your source doesn't define n, you might stop to think about where else you have see the same inequality, and why it would be used here. That may suggest an answer to your question.

Or, you might consider the fact that the t test can always be used; the z test is making an assumption that is only justified under certain conditions (and perhaps not really even then). Do they give you any restrictions on when you can use t?

It's true that I can't find a source that has n>30 and defines n. What I was taught in high school however is that Z-test is like T-test but n>30 (and that's the only definition we got). My current statistics professor didn't give any restrictions on when to use T-test except that it should only be used in finding whether or not there's a significant difference between two variables with quantitative data. Actually, she didn't teach about Z-test either, so I suppose it's safe to assume that she doesn't intend for us to use it for her project. Glad to hear that the T-test can always be used though. Thank you so much for your reply!

 

[Dr.Peterson]

It's true that I can't find a source that has n>30 and defines n. What I was taught in high school however is that Z-test is like T-test but n>30 (and that's the only definition we got). My current statistics professor didn't give any restrictions on when to use T-test except that it should only be used in finding whether or not there's a significant difference between two variables with quantitative data. Actually, she didn't teach about Z-test either, so I suppose it's safe to assume that she doesn't intend for us to use it for her project. Glad to hear that the T-test can always be used though. Thank you so much for your reply!

So, what is your source, and what does it say?

I expect that the n>30 rule of thumb is based on any one sample having 30 elements, so that data about the sample adequately approximates the corresponding data for the population; so the n in this case would apply to each sample; your problem doesn't meet this requirement. (I haven't taught statistics, so I'm not sure of the origin or validity of the rule.)

So, why do you consider using z if t is what you were taught?

 

[Mackattack]

So, what is your source, and what does it say?

I expect that the n>30 rule of thumb is based on any one sample having 30 elements, so that data about the sample adequately approximates the corresponding data for the population; so the n in this case would apply to each sample; your problem doesn't meet this requirement. (I haven't taught statistics, so I'm not sure of the origin or validity of the rule.)

So, why do you consider using z if t is what you were taught?

My source about Z-test was from one of my high school teachers. I no longer have the original photocopy of the lecture, but I still have the notes I took down. There wasn't much info except for having the "same purpose as T-test but n>30. When n>30, switch to Z."

My college blockmates also remember the same definition, which is why we wondered if we should switch to Z-test. That's all.

Thank you for your input on "...any one sample having 30 elements..."

 

[BigBeachBanana]

My source about Z-test was from one of my high school teachers. I no longer have the original photocopy of the lecture, but I still have the notes I took down. There wasn't much info except for having the "same purpose as T-test but n>30. When n>30, switch to Z."

My college blockmates also remember the same definition, which is why we wondered if we should switch to Z-test. That's all.

Thank you for your input on "...any one sample having 30 elements..."

If the two sample sizes are independent then both should have n>30 to implement the z-test, else use the t-test.
If the two sample sizes are paired, then each pair is counted as one sample, ergo you need >30 pairs for the z-test.

The t-test is often used when the population means and the standard deviation is not known. The t-test is often used for small samples because their distributions may not be normally distributed, while also accounts for additional variances due to limited sample size. If the sample is large (n>=30) then the Central Limit Theorem says that the sample mean is normally distributed and a z-test for a single mean can be used. In short, as the sample size n grows larger and larger, the results of a t-test and z-test become closer and closer. With infinite degrees of freedom, the results of the t and z tests become identical.

With that said, a t-test can also be used for larger sample sizes.

 

[Mackattack]

If the two sample sizes are independent then both should have n>30 to implement the z-test, else use the t-test.
If the two sample sizes are paired, then each pair is counted as one sample, ergo you need >30 pairs for the z-test.

The t-test is often used when the population means and the standard deviation is not known. The t-test is often used for small samples because their distributions may not be normally distributed, while also accounts for additional variances due to limited sample size. If the sample is large (n>=30) then the Central Limit Theorem says that the sample mean is normally distributed and a z-test for a single mean can be used. In short, as the sample size n grows larger and larger, the results of a t-test and z-test become closer and closer. With infinite degrees of freedom, the results of the t and z tests become identical.

With that said, a t-test can also be used for larger sample sizes.

Thank you so much!