The One-Sample t-Test
Statistical Inference Using Estimated Standard Errors: The One-Sample t Test
- We’ve covered the basics of hypothesis testing using the one-sample z-test.
- The one-sample z-test is appropriate when we know the standard deviation in the population.
- In actual research, we rarely know the population standard deviation, so we have to estimate the standard error of the mean from our sample data:
We use 
instead of 
- When we use an estimated standard error of the mean in the denominator, we’re now using a one-sample t-test instead of a z-test:
- The t-score gives the estimated number of standard errors that our 
differs from 
(the value given in the null hypothesis:
- But this causes a problem
- By using the estimated standard error, we can no longer use the normal distribution (& z-table) to find our critical values – there is a separate t distribution where we get our critical values.
- The t distribution is pretty similar to the normal distribution, but does differ especially when your sample size is small
- So, the major difference here in terms of our hypothesis-testing procedure is that we need to look up our critical values from the t distribution rather than the z (normal) distribution.
- Appendix D (p. 573) provides these – how to use the Appendix
- Look at the column corresponding to your alpha level (e.g., non-directional, 
)
- Look down the first column to find your degrees of freedom (N – 1)
- So, for df = 10, the critical t is 2.228 for a non-directional test using .
- (Notice at the bottom, when df is infinite, critical t is 1.96 – same as for z)
Steps of One-Sample t-test
1. State hypotheses:
- Non-directional (two-tailed) test
H0: 
H1: 
2. State decision rules:
- For df = N –1 = 15 – 1 = 14 and 
, critical t-value = 2.145
- If tobs > +2.145, or tobs < -2.145, reject null
- If –2.145 
tobs +2.145, do not reject null
3. Determine 
for sample size:
4. Calculate tobs:
5. Compare tobs to tcrit:
- -2.145 < -2.123 < +2.145 – do not reject null – the observed difference is most likely due to sampling error.
6. State the conclusion in words:
- The mean of 14 is not statistically significantly different from 15, so on average, students seem to be studying the recommended number of hours per week.
Confidence Intervals
- With the formal hypothesis-testing procedures we’ve covered, we tested a sample mean against a specific value of 
- Often we may not be very confident in the value of the true population mean being exactly the value we observed for our sample mean (because of sampling error).
as our best estimate of , and we specify a range of values around the that The formula looks like this:
From: 
To: 
Structure of the formula is the same, except that the t distribution is used in place of the z distribution and ŝ x is used in place of σx
From: 
To: 
where t is the nondirectional t critical value
CI especially provides more info when you reject the null because you get a range for the actual population mean for your observed mean (e.g., the actual population mean for our sample of SUNY students)
- With traditional hypothesis testing, we’d reject null because our 
falls beyond a critical value
- With the 95% CI around , we can likewise see that the hypothesized is not in the confidence interval.
