# Student's t-distribution

### Student's t-distribution

#### Lessons

In the previous section we discovered how to make a confidence interval for estimating population mean. However we knew what the population standard deviation ($\sigma$) was. However it is not always the case that $\sigma$ is known.

If population standard deviation ($\sigma$) is unknown then to make a confidence interval to estimate population mean we cannot our old formula for error: $E=Z_\frac{\sigma}{2}*\frac{\sigma}{\sqrt{n}}$ as it requires a knowledge of $\sigma$. So instead we are required to use a thing called t-scores ($t_{\frac{\alpha}{2}})$.

Once we find the t-scores for particular values (this is done in a similar way to finding z-scores) we have a new formula for the Margin of Error:
$E=Z_\frac{\sigma}{2}*\frac{S}{\sqrt{n}}$
• Introduction
How do we estimate population mean when ? is unknown?

• 1.
Determining a Confidence Interval for a Population Mean using t-distributions
The "Vendee Globe" is an around the world solo yacht race. In a particular year 31 sailors did the race and finished with an average time of 123 days, with a standard deviation of 11 days. With a t-score of $t_\frac{\alpha}{2}=2.45$ construct a confidence interval for the average amount of time it takes the average Vendee Globe sailor to circumnavigate the world (sail around the world).

• 2.
In "Anchiles", a small made-up town near the equator, 15 random days were sampled and found to have an average temperature of 28°C, with a standard deviation of 4°C. Assume that the average daily temperature of this town is normally distributed.
a)
With a 95% confidence where does the average daily temperature of Anchiles lie?

b)
What if we knew that in fact that the standard deviation of temperature was 4°C for the entire population? Then with a 95% confidence where does the average daily temperature of Anchiles lie?

c)
From the previous two questions, which has a larger confidence interval? Why might that be the case? Look at the t-scores as the sample gets larger and larger.

• 3.
Determining the Sample Standard Deviation with a given Margin of Error
From a sample of 25 new drivers it was found that the average age that a young adult in British Columbia receives their driver's license is given with a 90% confidence as somewhere in the interval of 16.72< $\mu$ <23.28 years old. Assume that the age that new drivers receive their license is normally distributed. What was the standard deviation from this sample?