Calculating t-score when population mean is unknown

[seven_percent]

In my stats class, we have a formulat for a single sample t-test which
is given as:

t = (Xbar - mu) / sigma

where: Xbar is the sample mean, mu is the population mean, and sigma
is the population StdDev.

It goes on to say that if sigma is unknown, the StdErr may be used
which is:

sigma(Xbar) = S / Sqrt(N)

where: S is the sample StdDev and N is the sample size.

What if the population mean is unknown also? How can I calculate the
t-score? My sample size is 10.

[Bruce Weaver]

seven_percent wrote:
> In my stats class, we have a formulat for a single sample t-test which
> is given as:
>
> t = (Xbar - mu) / sigma
>
> where: Xbar is the sample mean, mu is the population mean, and sigma
> is the population StdDev.

This is not correct. t = (Xbar-mu)/SE_Xbar where SE_Xbar = S/SQRT(n).

>
> It goes on to say that if sigma is unknown, the StdErr may be used
> which is:
>
> sigma(Xbar) = S / Sqrt(N)
>
> where: S is the sample StdDev and N is the sample size.

I think the distinction you are trying to make is between knowing and
not knowing the population standard deviation, sigma.

If sigma is known:

z = (Xbar - mu)/SE where SE = sigma/SQRT(n)

If sigma is NOT known, you use the sample standard deviation S as an
estimate of sigma:

t = (Xbar - mu)/SE where SE = S/SQRT(n)

n = sample size
S = sample standard deviation (with division by n-1)
Xbar = sample mean
mu = population mean given some null hypothesis
df for the t-ratio = n-1

> What if the population mean is unknown also? How can I calculate the
> t-score? My sample size is 10.

The value of mu is specified in some null hypothesis you wish to test.
So the only way it can be unknown is if you have no null hypothesis.

--
Bruce Weaver
bwe...@lakeheadu.ca
www.angelfire.com/wv/bwhomedir

[seven_percent]

I guess my teacher's notes are incorrect then. I am trying to apply
the concepts from the lesson to determine if $465,000 is a reasonable
amount to ask for our house. My "research hypothesis" in this case is
"I am 95% certain that $465,000 is a reasonable amount to ask for our
house." I have 10 home prices for the same # of bedrooms, bathrooms,
sqft, and lot size. I calculated the median, mode, midpoint, mean, and
standard deviation. I began to use a normal distribution and thought
that I could see what the z-score would be for $465,000. Then I
remembered that with sample sizes < 1000, you are supposed to use
t-scores instead. That's where I ran into problems. Any ideas for
this problem?