Question

the worlds smallest mammal is the bumblebee bat (also known as kittis hog - nosed bat or craseonycteris thonglongyai). such bats are roughly the size of a large bumblebee. a sample of bats, weighed in grams, is given below. assume that bat weights are normally distributed. find the 99% confidence interval of the mean.
do not round in between steps. round answers to at least 4 decimal places.

Explanation:

Step1: Calculate the sample mean $\bar{x}$

The formula for the sample mean is $\bar{x}=\frac{\sum_{i = 1}^{n}x_{i}}{n}$.
Here, $n = 10$, and $\sum_{i=1}^{10}x_{i}=2.03 + 1.53+1.58 + 1.9+1.71+2.02+1.63+2.23+1.66+2.27=18.59$.
So, $\bar{x}=\frac{18.59}{10}=1.859$.

Step2: Calculate the sample standard deviation $s$

The formula for the sample standard deviation is $s=\sqrt{\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}{n - 1}}$.
First, calculate $(x_{i}-\bar{x})^{2}$ for each $x_{i}$:
$(2.03 - 1.859)^{2}=0.029241$, $(1.53-1.859)^{2}=0.108241$, $(1.58 - 1.859)^{2}=0.077841$, $(1.9-1.859)^{2}=0.001681$, $(1.71-1.859)^{2}=0.022201$, $(2.02-1.859)^{2}=0.025921$, $(1.63-1.859)^{2}=0.052441$, $(2.23-1.859)^{2}=0.137641$, $(1.66-1.859)^{2}=0.039601$, $(2.27-1.859)^{2}=0.168921$.
$\sum_{i = 1}^{10}(x_{i}-\bar{x})^{2}=0.029241+0.108241+0.077841+0.001681+0.022201+0.025921+0.052441+0.137641+0.039601+0.168921 = 0.66373$.
Then $s=\sqrt{\frac{0.66373}{9}}\approx0.2716$.

Step3: Determine the critical value $t_{\alpha/2}$

The confidence level is $99\%$, so $\alpha=1 - 0.99 = 0.01$, and $\frac{\alpha}{2}=0.005$.
The degrees of freedom $df=n - 1=9$.
Using the $t$-distribution table or a calculator, $t_{0.005,9}=3.2498$.

Step4: Calculate the margin of error $E$

The formula for the margin of error for a confidence interval of the mean (when population standard deviation $\sigma$ is unknown) is $E = t_{\alpha/2}\frac{s}{\sqrt{n}}$.
Substitute $t_{\alpha/2}=3.2498$, $s = 0.2716$, and $n = 10$ into the formula:
$E=3.2498\times\frac{0.2716}{\sqrt{10}}\approx3.2498\times0.0859\approx0.2792$.

Step5: Calculate the confidence interval

The confidence interval is $\bar{x}-E<\mu<\bar{x}+E$.
Substitute $\bar{x}=1.859$ and $E = 0.2792$:
$1.859-0.2792<\mu<1.859 + 0.2792$.

Answer:

$1.5798<\mu<2.1382$