Question

golf cart attendants at a local country club often receive tips from golfers they assist with golf clubs. to help recruit new employees, the country club would like to determine the average tips per day an attendant earns. a sample of 17 attendants averaged $35.04 per day in tips. assume the population standard deviation for daily tips is $9.32. complete parts a and b below.
a. construct a 95% confidence interval to estimate the average daily tips for an attendant.
the average daily tips for an attendant is between a lower limit of $ and an upper limit of $ (round to the nearest cent as needed.)

Explanation:

Step1: Identify the formula

For a 95% confidence interval when the population standard - deviation $\sigma$ is known, the formula is $\bar{x}\pm z_{\alpha/2}\frac{\sigma}{\sqrt{n}}$, where $\bar{x}$ is the sample mean, $z_{\alpha/2}$ is the z - score, $\sigma$ is the population standard deviation, and $n$ is the sample size. For a 95% confidence interval, $\alpha = 1 - 0.95=0.05$, so $\alpha/2 = 0.025$, and $z_{\alpha/2}=1.96$. Given $\bar{x}=35.04$, $\sigma = 9.32$, and $n = 17$.

Step2: Calculate the margin of error $E$

$E = z_{\alpha/2}\frac{\sigma}{\sqrt{n}}=1.96\times\frac{9.32}{\sqrt{17}}$. First, calculate $\sqrt{17}\approx4.1231$. Then $\frac{9.32}{4.1231}\approx2.26$. And $1.96\times2.26 = 4.43$.

Step3: Calculate the lower and upper limits

The lower limit $L=\bar{x}-E=35.04 - 4.43=30.61$. The upper limit $U=\bar{x}+E=35.04 + 4.43=39.47$.

Answer:

The average daily tips for an attendant is between a lower limit of $\$30.61$ and an upper limit of $\$39.47$.