3.3.4 One sample $t$ confidence intervals

Dolphins are at the top of the oceanic food chain, which causes dangerous substances such as mercury to concentrate in their organs and muscles. This is an important problem for both dolphins and other animals, like humans, who occasionally eat them. For instance, this is particularly relevant in Japan where school meals have included dolphin at times.

Here we identify a confidence interval for the average mercury content in dolphin muscle using a sample of 19 Risso’s dolphins from the Taiji area in Japan.³³³³Taiji is a significant source of dolphin and whale meat in Japan. Thousands of dolphins pass through the Taiji area annually, and we will assume these 19 dolphins represent a simple random sample from those dolphins. Data reference: Endo T and Haraguchi K. 2009. High mercury levels in hair samples from residents of Taiji, a Japanese whaling town. Marine Pollution Bulletin 60(5):743-747. The data are summarized in Table 3.6. The minimum and maximum observed values can be used to evaluate whether or not there are obvious outliers or skew.

Answer. The observations are a simple random sample and consist of less than 10% of the population, therefore independence is reasonable. The summary statistics in Table 3.6 do not suggest any skew or outliers; all observations are within 2.5 standard deviations of the mean. Based on this evidence, the normality assumption seems reasonable. In the normal model, we used $z^{\star }$ and the standard error to determine the width of a confidence interval. We revise the confidence interval formula slightly when using the $t$ distribution:

The sample mean and estimated standard error are computed just as before ($\overline{x}=4.4$ and $SE=s/\sqrt{n}=0.528$). The value $t_{df}^{\star }$ is a cutoff we obtain based on the confidence level and the $t$ distribution with $df$ degrees of freedom. Before determining this cutoff, we will first need the degrees of freedom.

In our current example, we should use the $t$ distribution with $df=19-1=18$ degrees of freedom. Then identifying $t_{18}^{\star }$ is similar to how we found $z^{\star }$.

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  For a 95% confidence interval, we want to find the cutoff $t_{18}^{\star }$ such that 95% of the $t$ distribution is between -$t_{18}^{\star }$ and $t_{18}^{\star }$.

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  We want hence we look this up in R: qt(0.975,df=18)=2.100922. Note we use 97.5% as we want a two-tailed answer so 2.5% in the upper and lower tails.

Generally the value of $t_{df}^{\star }$ is slightly larger than what we would get under the normal model with $z^{\star }$.

Finally, we can substitute all our values into the confidence interval equation to create the 95% confidence interval for the average mercury content in muscles from Risso’s dolphins that pass through the Taiji area:

We are 95% confident the average mercury content of muscles in Risso’s dolphins is between 3.29 and 5.51 $\mu$g/wet gram. This is above the Japanese regulation level of 0.4 $\mu$g/wet gram.

Answer. There are no obvious outliers; all observations are within 2 standard deviations of the mean. If there is skew, it is not evident. There are no red flags for the normal model based on this (limited) information, and we do not have reason to believe the mercury content is not nearly normal in this type of fish.

Answer. The standard error: $SE=\frac{0.069}{\sqrt{15}}=0.0178$. Degrees of freedom: $df=n-1=14$.

Looking in the table in the Appendix for t-distributions in the $df=14$ row for the value 0.05 (for a 90% confidence interval - two tails) we identify thus $t_{14}^{\star }=1.76$ (recall the t-distribution is symmetric).

R> qt(0.95,df=14)

Answer. $\overline{x}\pm t_{14}^{\star }SE\to \mathrm{\hspace{0.25em}0.287}\pm \mathrm{\hspace{0.25em}1.76}\times 0.0178\to (0.256,0.318)$. We are 90% confident that the average mercury content of croaker white fish (Pacific) is between 0.256 and 0.318 ppm.