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 (σ\sigma) was. However it is not always the case that σ\sigma is known.

If population standard deviation (σ\sigma) is unknown then to make a confidence interval to estimate population mean we cannot our old formula for error: E=Zσ2σnE=Z_\frac{\sigma}{2}*\frac{\sigma}{\sqrt{n}} as it requires a knowledge of σ\sigma. So instead we are required to use a thing called t-scores (tα2)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=Zσ2SnE=Z_\frac{\sigma}{2}*\frac{S}{\sqrt{n}}
  • 1.
    How do we estimate population mean when ? is unknown?

  • 2.
    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 tα2=2.45t_\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).

  • 3.
    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.


  • 4.
    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< μ\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?