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 $35.94 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 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.

Step2: Determine the values

We are given that $\bar{x} = 35.94$, $\sigma=9.32$, $n = 17$. For a 95% confidence interval, $\alpha=1 - 0.95=0.05$, so $\alpha/2=0.025$. The $z$ - score $z_{\alpha/2}=z_{0.025}=1.96$.

Step3: Calculate the margin of error

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

Step4: Calculate the confidence interval

The lower limit is $\bar{x}-E=35.94 - 4.43=31.51$.
The upper limit is $\bar{x}+E=35.94 + 4.43=40.37$.

Answer:

The average daily tips for an attendant is between a lower limit of $\$31.51$ and an upper limit of $\$40.37$.