Question

Question 1. A simple random sample survey of a large car dealership collects information on the average hours worked and the average amount of commission collected by employees. There are 112 employees and the overall average number of hours worked is 21. The data table is given below.

Part i) Build a 95% percent confidence interval for the mean commission and the total commission

Part ii) Using ratio estimation, what should the minimum sample size be to be accurate to within 1 unit 19 times out of 20?

Data Table:

Unit	Commission	Hours_Worked
1	160	12
2	242	22
3	274	24
4	373	18
5	517	31
6	120	24
7	161	16
8	230	18
9	741	33
10	646	32
11	367	21
12	294	19
13	197	14
14	192	13
15	173	6
16	182	14
17	146	17
18	590	25
19	240	29
20	126	8
21	412	27
22	275	28
23	153	8
24	213	26
25	204	19
26	248	14
27	911	34
28	200	35
29	609	26
30	294	19
31	127	26
32	378	21
33	383	30
34	231	31
35	239	24
36	188	30
37	180	18
38	418	30

Answer 1

Calculating Mean Commission CI

Statistics

Variable N   N*   Mean   SE Mean   StDev   Variance   Minimum   Q1   Median   Q3   Maximum
Commission   38   0 306.2 30.0 185.2 34306.4 120.0   181.5   239.5   379.3   911.0

Step 1: start with

• the number of observations n
• the mean x
• and the standard deviation s

Note: we should use the standard deviation of the entire population, but in many cases we won't know it.

We can use the standard deviation for the sample if we have enough observations (at least n=30, hopefully more).

Using our example:

• number of observations n = 38
• mean x = 306.2
• standard deviation s = 185.2

Step 2: decide what Confidence Interval we want: 95% or 99% are common choices. Then find the "Z" value for that Confidence Interval here:

Confidence Interval	Z
95%	1.960

For 95% the Z value is 1.960

Step 3: use that Z value in this formula for the Confidence Interval

x ± Zs√n

Where:

• x is the mean of the mean commission
• Z is the chosen Z-value from the table above
• s is the standard deviation of the mean commission
• n is the number of sample observations

And we have:

306.2 ± 1.960 × 185.2/√38

Which is:

306.2 ± 58.92

Calculating Total Commission CI

X ± ZS√N

Where:

• X is the mean of the Total Commission
• Z is the chosen Z-value from the table above
• s is the standard deviation of the Total Commission
• N is the number of obs

Using our example:

• number of observations N = 112
• mean X = 7579  
• standard deviation S = 6697  

And we have:

7579 ± 1.960 × 6697/√112

Which is:

7579 ± 1240.654

Descriptive Statistics: total Commission
Statistics

Variable N   N*   Mean   SE Mean   StDev   Variance   Minimum   Q1   Median   Q3
total Commission   38   0   7579 1086   6697   44849551   1008   2850   5613   8735
Variable   Maximum
total Commission   30974
b. Ration Estimator