Question

a cruise company would like to estimate the average beer consumption to plan its beer inventory levels on future seve (certainly doesnt want to run out of beer in the middle of the ocean!) the average beer consumption over 18 randomly 81,706 bottles with a sample standard deviation of 4,573 bottles. complete parts a and b below.
a. construct a 99% confidence interval to estimate the average beer consumption per cruise.
the 99% confidence interval to estimate the average beer consumption per cruise is from a lower limit of 78,583 bottles (round to the nearest whole numbers.)
b. what assumptions need to be made about this population?
a. the only assumption needed is that the population follows the students t - distribution.
b. the only assumption needed is that the population d,tribution is skewed to one side.
c. the only assumption needed is that the population size is larger than 30.
d. the only assumption needed is that the population follows the normal probability distribution.

Explanation:

Step1: Identify the formula for confidence interval

For a small - sample (n < 30) when population standard deviation is unknown, the confidence interval for the population mean $\mu$ is given by $\bar{x}\pm t_{\alpha/2}\frac{s}{\sqrt{n}}$, where $\bar{x}$ is the sample mean, $s$ is the sample standard deviation, $n$ is the sample size, and $t_{\alpha/2}$ is the critical value from the t - distribution. Here, $n = 18$, $\bar{x}=81706$, $s = 4573$. For a 99% confidence interval, $\alpha=1 - 0.99=0.01$, and $\alpha/2=0.005$. The degrees of freedom $df=n - 1=18 - 1 = 17$.

Step2: Find the critical value $t_{\alpha/2}$

Looking up in the t - distribution table, $t_{0.005,17}=2.898$.

Step3: Calculate the margin of error $E$

$E=t_{\alpha/2}\frac{s}{\sqrt{n}}=2.898\times\frac{4573}{\sqrt{18}}\approx2.898\times\frac{4573}{4.243}\approx2.898\times1077.8=3123$.

Step4: Calculate the lower and upper limits

The lower limit is $\bar{x}-E=81706 - 3123=78583$. The upper limit is $\bar{x}+E=81706+3123 = 84829$.

For part b:
When constructing a confidence interval for the population mean using the t - distribution (since the population standard deviation is unknown and the sample size is small), the main assumption is that the population from which the sample is drawn follows a normal probability distribution.

Answer:

a. The 99% confidence interval is from 78,583 bottles to 84,829 bottles.
b. D. The only assumption needed is that the population follows the normal probability distribution.