Question

a cruise company would like to estimate the average beer consumption to plan its beer inventory levels on future seven - day cruises. (the ship certainly doesnt want to run out of beer in the middle of the ocean!) the average beer consumption over 18 randomly selected seven - day cruises was 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 bottles to an upper limit of bottles.
(round to the nearest whole numbers.)

Explanation:

Step1: Identify the formula

The formula for a confidence interval when the population standard - deviation is unknown is $\bar{x}\pm t_{\alpha/2}\frac{s}{\sqrt{n}}$, where $\bar{x}$ is the sample mean, $t_{\alpha/2}$ is the critical value, $s$ is the sample standard deviation, and $n$ is the sample size.

Step2: Determine the values of $\bar{x}$, $s$, and $n$

We are given that $\bar{x} = 81706$, $s = 4573$, and $n = 18$.

Step3: Find the degrees of freedom and $t_{\alpha/2}$ value

The degrees of freedom $df=n - 1=18 - 1 = 17$. For a 99% confidence interval, $\alpha=1 - 0.99 = 0.01$, and $\alpha/2=0.005$. Looking up in the $t$-distribution table, $t_{0.005,17}=2.898$.

Step4: Calculate the margin of error

The margin of error $E=t_{\alpha/2}\frac{s}{\sqrt{n}}=2.898\times\frac{4573}{\sqrt{18}}$.
First, calculate $\sqrt{18}\approx4.243$. Then, $\frac{4573}{4.243}\approx1077.8$. And $E = 2.898\times1077.8\approx3123.5$.

Step5: Calculate the confidence interval

The lower limit is $\bar{x}-E=81706 - 3123.5=78582.5\approx78583$.
The upper limit is $\bar{x}+E=81706+3123.5 = 84829.5\approx84830$.

Answer:

The 99% confidence interval to estimate the average beer consumption per cruise is from a lower limit of 78583 bottles to an upper limit of 84830 bottles.