Question

The file Assign1_Data contains data on the hourly wages (Hourly Wage) of all employees of a grocery chain in the Ottawa region.

1. (a) Treating this set of data as the population, use Minitab to calculate the population mean and the population standard deviation for the Hourly Wage variable.

2. (b) Examine a boxplot and histogram of the population data. Explain if the means of all possible random samples of size 40 from this population would form a normal distribution.

3. (c) Using your results above, estimate the mean and standard deviation of the sampling distribution model for the sample mean of the wage variable. Assume a sample size of 40.

Data:

Hourly Wage
15.0999999
15.24000001
15
18
17.30000019
20.75
17
15.5999999
18.25
20.13000011
20.77000046
17.5
19.5
15.5999999
16.5
18.88000011
20.47999954
18.32999992
12.52999997
18
21.56000042
19.78000021
15.25
16.5
21.68000031
17
16.67999983
16.26999998
18.1500001
15.50999999
15
18.25
19.80999994
16.5
16
18.38000011
13.66999996
14.93000007
15.6500001
14.9000001
13.63
20.60000038
17
18
14.5
15.25
15.4000001
16.38000011
18.25
15.71000004
19.78000021
18.25
17.71000004
14
17.71000004
16.90999985
14.91000009
15.75
16
15.0999999
20.44999981
19.13999987
16.5
16.6500001
14.9000001
18.67000008
15.5
15.25999999
15.25
20
21.85000038
19.5
17.90999985
15
16.80999994
18.5
16
15.5
16.25
15.5
17.13000011
15.75
16.5
19.63000011
18.8499999
18.67000008
14.52999997
21.80000019
15.36999989
17.4000001
18.11000013
16.19999981
15.75
15.5
15.6400001
15.79999995
15
17
16.63000011
15
15.20000005
15.91000009
18.42999983
17.48000002
13.5
14.9000001
17
20.92000008
17
15.51999998
14.9000001
16.5
14.25
17
15.75
19.9000001
16.71999979
17.84000015
15.82999992
15.20000005
14
16.5
14.1400001
14.38000011
15.75
17.51999998
18.5
15.0999999
18.63000011
14.30999994
18.88000011
14.82999992
15.13000011
20
16.5
20.64999962
14
16.75
18.25
18
17.25
14.17000008
19.13999987
18.21999979
21
17.76999998
16
20.75
18.53000021
19.5999999
17
17
20.64000034
15.29999995
16.44000006
16.55000019
15.5
18.25
15.8499999
18.17999983
14.91000009
18.25
18.25
21.05000019
18.88000011
20.75
15.04999995
15
17.80000019
16.0999999
20
18.1500001
14.70000005
14.75
15
15
19.36000013
19.5
15.5
20.10000038
15.75
15.25
17.82999992
15.5
15.32999992
16
15.5
18.25
14.95000005
17.71000004
15
21
20.32999992
15
17.75
18.76000023
15
15.5
15.25
16
14.92000008
15.05999994
15.20000005
16.75
15
15.5
16.11000013
13.96000004
16.28999996
15
18.44999981
17.19999981
16.5
15.88000011
15.45000005
16.0999999
15
17.9000001
16
15
15.54999995
15
20.75
14.9000001
18.26000023
15.5
16.5999999
18
14.8900001
17.57999992
16
18
16.5
14.92000008
16.32999992
16.28000021
16.57000017
18.25
14.95000005
20.75
20.5
15.75
15.1500001
17
18.46000004
14
16.78999996
17.78000021
15.18000007
16.67999983
16.0999999
14.91000009
18
15.5999999
15.95000005
19
15
18.07999992
20.63000011
15
15.75
14.9000001
15
18.25
15.5
15
15.24000001
20.02000046
15.32999992
17.25
18.25
15.5
14.95000005
15
16.69000006
15.73000002
16
16
14.9000001
15.04999995
17.05000019
18.25
17.25
16.78999996
15.3499999
15
20.43000031
17.69999981
15.5
16.23999977
19
18
16.5
15
14.9000001
16
17.25
16
15.29999995
17.05000019
15.57999992
17
16.57000017
15.45000005
16.63000011
14.92000008
16.51000023
18.5
19.5
15.53999996
16.19999981
15.50999999
16.5
15.3499999
14.91000009
17.25
16.05000019
15.75
15.4000001
15
18.28999996
14.53999996
16.5
15.13000011
18.36000013
16.67999983
18.80000019
20.52999973
16.17000008
15.75
15.25999999
21.13000011
16.5
15
20.75
16.13999987
14.86999989
15.3499999
18.07999992
15
16.19999981
17.5999999
15.75999999
15.0999999
16.28999996
19.5
16.05000019
16.6500001
17
14.9000001
20
20.43000031
14.92000008
18.25
18.25
17.11000013
16
16.44000006
18.88000011
17.42999983
15
14.9000001
18.25
16.34000015
15.25
19.26000023
18.3499999
17.63000011
20.75
15.20000005
15
15
14.88000011
15.3499999
18.5
16.5
15.80999994
20.80000019
21.42000008
18.32999992
16
14.9000001
15.5
18
18.36000013
15.54999995
15
16.5
18.63000011
21.30000019
15
15.25
13.5
17.9000001
20
14.9000001
15.28999996
18.5
16
18
16.07999992
15.75
15.04999995
15.5
14.92000008
16.5
15.3499999
17.94999981
20
15
17
17.5
14.6500001
15
16.5
20.18000031
21.09000015
15.25
16.5
16.5
15.71000004
18.5
14.9000001
17.5999999
14.23000002
17
20.32999992
14.9000001
18.25
16.55000019
15.27999997
14.29999995
15.29999995
15.1500001
17.1500001
15.13000011
19.25
14.9000001
13.75
14.8900001
14.9000001
18.25
14.5999999
18.63000011
15.5
18.5
15
16.38000011
16.94999981
21
13.42999995
15.07999992
21.32999992
19.5
16.75
17.6500001
14.26999998
16.67000008
15.5

Answer 1

a) Population mean=16.774, population SD=1.906 (divisor=N=459)

b) The histogram shows higher clustering of lower values. Thus the distribution looks like that of a positively skewed distribution.

From the boxplot, we observe that the central line inside the box(i.e. median) is closer to the lower end of the box(i.e. Q1) than to the upper end of the box(i.e. Q3). Hence the distribution deviates from symmetry. Thus the distribution does not appear to be normal.

Although  the parent distribution is not normal, but the sample size (n=40) for the calculation of means is quite large (as n>30) and hence by central limit law, the mean of all possible random samples of size 40 from this population would form a normal distribution.

c) If and are the population mean and SD, the estimates of the the mean and standard deviation of the sampling distribution model for the sample mean of the wage variable will be and with n=40.  Hence, the estimates of the the mean and standard deviation of the sampling distribution model for the sample mean of the wage variable are  16.774 and 0.3014.

For further query in above, comment.