Confusion between z and t values

[Rifscape]

Homework Statement

Hi,

Alright so I have some confusion on when to use specific tests and the z vs t test.

Given this example (not my homework) could someone please clarify.

Alright say you have a random sample of size 200. You find the sample mean to be 10 and the sample standard deviation to be 5.

What would be the answer to these questions?

What do you think the population sd and mean are?

1. Construct a 95% confidence interval for mu given sigma is know
2. Construct a 95% confidence interval for mu given sigma is unknown using s(sample sd)
   

First major question, what is the difference between 2 and 3? My reasoning is that for 2 you use the z alpha/2 value and for 3 you use the t alpha/2 value, is this correct?

1. Assume you would like to have your estimate to be with 0.5 units from µ with a 95% CI, what should the sample size be? For this question should I use the z value or t value? I know how to calculate after that.

Ignore the hypothesis values of 3 and 2, again I am unsure whether to use a t or z value here.

5.Please construct a test statistic to test H0 : µ ≤ 3 versus Ha : µ > 3, and provide the formula for the test statistic. At a probability of Type I error α = .05, do you reject H0? Why or why not?

Consider H0 : µ = 2 versus Ha : µ != 2. At a probability of Type I error α = .05, do you reject H0?

Why or why not?Please construct a test statistic to test H0 : µ ≤ 3 versus Ha : µ > 3, and provide the formula for the test statistic. At a probability of Type I error α = .05, do you reject H0? Why or why not? Consider H0 : µ = 2 versus Ha : µ != 2. At a probability of Type I error α = .05, do you reject H0? Why or why not?

This is bonus but I would be happy to hear if you know something about it

1. For testing H0 : µ = 2 versus Ha : µ != 2, if s is smaller than σ and the test statistic is based on s instead of on σ, is it easier to reject H0? Why or why not?

Overall I am confused on when to use z vs t and the conceptual notions behind it. Any help is appreciated.

Homework Equations

The Z and T value equations, my question is more conceptual though.

The Attempt at a Solution

I outlined some explanation above. I apologize for the poor formatting, and will reformat if needed.

Thank you for your time

 

[Ray Vickson]

Homework Statement

Hi,

Alright so I have some confusion on when to use specific tests and the z vs t test.

Given this example (not my homework) could someone please clarify.

Alright say you have a random sample of size 200. You find the sample mean to be 10 and the sample standard deviation to be 5.

What would be the answer to these questions?

What do you think the population sd and mean are?

1. Construct a 95% confidence interval for mu given sigma is know
2. Construct a 95% confidence interval for mu given sigma is unknown using s(sample sd)

First major question, what is the difference between 2 and 3? My reasoning is that for 2 you use the z alpha/2 value and for 3 you use the t alpha/2 value, is this correct?

1. Assume you would like to have your estimate to be with 0.5 units from µ with a 95% CI, what should the sample size be? For this question should I use the z value or t value? I know how to calculate after that.

Ignore the hypothesis values of 3 and 2, again I am unsure whether to use a t or z value here.

5.Please construct a test statistic to test H0 : µ ≤ 3 versus Ha : µ > 3, and provide the formula for the test statistic. At a probability of Type I error α = .05, do you reject H0? Why or why not?

Consider H0 : µ = 2 versus Ha : µ != 2. At a probability of Type I error α = .05, do you reject H0?

Why or why not?Please construct a test statistic to test H0 : µ ≤ 3 versus Ha : µ > 3, and provide the formula for the test statistic. At a probability of Type I error α = .05, do you reject H0? Why or why not? Consider H0 : µ = 2 versus Ha : µ != 2. At a probability of Type I error α = .05, do you reject H0? Why or why not?

This is bonus but I would be happy to hear if you know something about it

1. For testing H0 : µ = 2 versus Ha : µ != 2, if s is smaller than σ and the test statistic is based on s instead of on σ, is it easier to reject H0? Why or why not?

Overall I am confused on when to use z vs t and the conceptual notions behind it. Any help is appreciated.

Homework Equations

The Z and T value equations, my question is more conceptual though.

The Attempt at a Solution

I outlined some explanation above. I apologize for the poor formatting, and will reformat if needed.

Thank you for your time

For a normal population:
(1) Use normal distribution (i.e., points $z$) for inferences about the mean when you know the variance.
(2) Use the t-distribution to make inferences about the mean when the variance is not known but is estimated from the same data you are analyzing.

When the sample size $n$ is large enough there is not much difference between the $T(n-1)$ and $Z$. However, the differences are important for small to moderate values of $n$. You can do an internet search to get information on what the appropriate $n$ ranges should be.

Now you ought to be able to answer all the questions on your own, and PF rules require you to make such an effort.

 

[Rifscape]

Ray you save me every time haha.

Yeah I understand how to use the formulas to answer it. So based on the information you gave me, I would only use the Z critical value for the question which states that you know the population standard deviation. However for all other parts I would use the T value, since we are only given a random sample of size n > 30 for which we can calculate the sample standard deviation and sample mean. Since the sample size is greater than 30 I can use the sample mean as an estimate of the population mean, but will have to use the T value since I am using the sample standard deviation without knowing the population standard deviation. Is this reasoning correct?

Thanks again

 

[Ray Vickson]

Ray you save me every time haha.

Yeah I understand how to use the formulas to answer it. So based on the information you gave me, I would only use the Z critical value for the question which states that you know the population standard deviation. However for all other parts I would use the T value, since we are only given a random sample of size n > 30 for which we can calculate the sample standard deviation and sample mean. Since the sample size is greater than 30 I can use the sample mean as an estimate of the population mean, but will have to use the T value since I am using the sample standard deviation without knowing the population standard deviation. Is this reasoning correct?

Thanks again

Basically, that is what I said, I think.

 

[Rifscape]

Alright cool that makes sense, one quick question though. After consulting the textbook, it seems that when you calculate sample size you always use the z score value even when you have to use s instead of sigma, is there a reason for this?

When I used the z score value in the sample size equation I get around 13~14.

 

[Ray Vickson]

Alright cool that makes sense, one quick question though. After consulting the textbook, it seems that when you calculate sample size you always use the z score value even when you have to use s instead of sigma, is there a reason for this?

When I used the z score value in the sample size equation I get around 13~14.

It is impossible to compute a z-score if you do not know $\sigma$. If you do, in fact, know $\sigma$ the best policy is to use that and get a true (normally-distributed) z-score; the reason that is a better policy---if you can do it---is that the normal distribution is "narrower" than the t-distribution, so you get stronger statistical statements---narrower confidence intervals, etc. If you do not know $\sigma$ all you can do is replace $\sigma$ by $s$ (the sample standard deviation); then it is not really a z-score anymore, but rather, is a t-value. However, for all I know your un-named textbook may continue to denote the quantity by $z$, but that would just be notation and not anything fundamental.