Question

A study was conducted to test whether a manufacturing process performed at a remote location can be established locally. Test devices (pilots) were set up at both the old and new locations and voltage readings on 30 production runs at each location were obtained. The data are reproduced in the table below:
VOLTAGE
Old Location New Location
9.98 10.12 9.84 9.19 10.01 8.82
10.26 10.05 10.15 9.63 8.82 8.65
10.05 9.80 10.02 10.10 9.43 8.51
10.29 10.15 9.80 9.70 10.03 9.14
10.03 10.00 9.73 10.09 9.85 9.75
8.05 9.87 10.01 9.60 9.27 8.78 10.55 9.55 9.98 10.05 8.83 9.35
10.26 9.95 8.72 10.12 9.39 9.54
9.97 9.70 8.80 9.49 9.48 9.36
9.87 8.72 9.84 9.37 9.64 8.68

Required:
a. Construct a relative frequency histogram for the voltage readings of the old process.
b. Construct a stem-and-leaf display for the voltage readings of the old process. Which of the two graphs in parts a and b is more informative about where most of the voltage readings lie?
c. Construct a relative frequency histogram for the voltage readings of the new process.
d. Compare the two graphs in parts a and c. (You may want to draw the two histograms on the same graph.) Does it appear that the manufacturing process can be established locally (i.e., is the new process as good as or better than the old)?
e. Find and interpret the mean, median, and mode for each of the voltage readings data sets. Which is the preferred measure of central tendency? Explain.
f. Calculate the z-score for a voltage reading of 10.50 at the old location.
g. Calculate the z-score for a voltage reading of 10.50 at the new location.
h. Based on the results of parts f and g, at which location is a voltage reading of 10.50 more likely to occur? Explain.
i. Construct a box plot for the data at the old location. Do you detect any outliers?
j. Use the method of z-scores to detect outliers at the old location.
k. Construct a box plot for the data at the new location. Do you detect any outliers?
l. Use the method of z-scores to detect outliers at the new location.
m. Compare the distributions of voltage readings at the two locations by placing the box plots, parts i and k, side by side vertically.

Answer 1

Answer:

Step-by-step explanation:

Hello!

You have the information about voltage readings at an old and a new manufacturing location obtained remotely.

a, b and c in the attachment.

Histograms for a and c:

To construct a frequency histogram you have to first arrange the data for both locations in a frequency table. For this, I'm going to determine 5 class interval for each location. To do so you need to calculate the width of the intervals. First, you calculate the range of the variable and then you have to divide it by the number of intervals you want to do.

Old location: Range= 10,55-8,05= 2,5 → Class width: 2,5/5= 0,5

New Location: Range= 10,12-8,51= 1,61 → Class width: 1,61/5= 0,322

Starting from the minimum value you add the calculated width and create the intervals:

Old Location:

8,05-8,55

8,55-9,05

9,05-9,55

9,55-10,05

10,05-10,55

New Location

8,51-8,83

8,83-9,15

9,15-9,48

9,48-9,80

9,80-10,12

Stem and Leaf diagram for b:

To construct this diagram first I've ordered the data from leat to greatests. Then I've used the integer to form the stem 8,- 9.- and 10.- and the decimals are placed in the leafs of the diagram.

Comparing it to the histogram and stem and leaf diagram for the readings of the Old Location, the histogram stem, and leaf diagram show better where most of the readings lie.

d.

Comparing both histograms, it looks like the readings in the new location are more variable than the readings in the old location but more uniformly distributed. I would say that the readings in the new location are better than the readings in the old location.

e.

To calculate the mean you have to apply the following formula:

X[bar]= (∑xi'fi)/n

X[bar]OLD=(∑xi'fi)/n= (8.3*1+8.8*3+9.3*0+9.8*17+10.3*9)/30= 294/30= 9.8

X[bar]NEW=(∑xi'fi)/n= (8.67*6+8.99*2+9.315*7+9.64*8+9.96*7)/30= 282.045/30= 9.4015≅9.40

First you have to calculate the position of the median:

For both data sets the PosMe= 30/2=15

Now you arrange the data from least to highest and determine wich observation is in the 15th position:

Old Location

8,05 , 8,72 , 8,72 , 8,8 , 9,55 , 9,7 , 9,73 , 9,8 , 9,8 , 9,84 , 9,84 , 9,87 , 9,87 , 9,95 , 9,97 , 9,98 , 9,98 , 10 , 10,01 , 10,02 , 10,03 , 10,05 , 10,05 , 10,12 , 10,15 , 10,15 , 10,26 , 10,26 , 10,29 , 10,55

MeOLD= 9.97

New Location

8,51 , 8,65 , 8,68 , 8,78 , 8,82 , 8,82 , 8,83 , 9,14 , 9,19 , 9,27 , 9,35 , 9,36 , 9,37 , 9,39 , 9,43 , 9,48 , 9,49 , 9,54 , 9,6 , 9,63 , 9,64 , 9,7 , 9,75 , 9,85 , 10,01 , 10,03 , 10,05 , 10,09 , 10,1 , 10,12

MeNEW= 9.43

The mode is the observation with more absolute frequency.

To determine the mode on both data sets I'll use the followinf formula:

Md= Li + c [Δ₁/(Δ₁+Δ₂)]

Li= Lower bond of the interval with most absolute frequency (modal interval)

c= amplitude of the modal interval

Δ₁= absolute frequency of the modal interval minus abolute frequency of the previous interval

Δ₂= absolute frequency of the modal interval minus the absolute frequency of the next interval

Modal interval OLD

9,55-10,05

Δ₁= 17-0= 17

Δ₂= 17-9= 8

c= 0.5

Li= 9.55

MdOLD= 9.55 + 0.5*[17/(17+8)]= 9.89

Modal interval NEW

9,48-9,80

Δ₁= 8-7= 1

Δ₂= 8-7= 1

c= 0.32

Li= 9.48

MdNEW= 9.48+0.32*[1/(1+1)]= 9.64

f.

OLD

Mean 9.8

SE 0.45

X= 10.50

Z= (10.50-9.8)/0.45= 1.56

g.

NEW

Mean 9.4

SE 0.48

X=10.50

Z= (10.50-9.4)/0.48=2.29

h. The Z score for the reading 10.50 for the old location is less than the Z score for the reading 10.50 for the new location, this means that the reading is closer to the mean in the old location than in the new location.

The reading 10.50 is more unusual for the new location.

i. and k. Boxplots attached.

There are outliers for the readings in the old location, none in the readings for the new location.

j. To detect outliers using the Z- score you have to "standardize every value of the data set using the corresponding values of the mean and standard deviation. Observations that obtained a Z-score greater than 3 or less than -3 are outliers.

The data set for the new location has no outliers, To prove it I've calculated the Z-scores for the max and min values:

Min: Z=(8.51-9.4)/0.48= -1.85

Max: Z=(10.12-9.4)/0.48= 1.5

The records for the old location show, as seen in the boxplot, outliers:

To find them I'll start calculating values of Z from the bottom and the top of the list until getting a value Z≥-3 and Z≤3

Bottom:

1) 8,05 ⇒ Z=(8.05-9.8)/0.45= -3.89

2) 8,72 ⇒ Z= (8.72-9.8)/0.45= -2.4

Top

1) 10,55⇒ Z= (10.55-9.8)/0.45= 1.67

m. As mentioned before, the distribution for the new location seems to be more uniform and better distributed than the distribution for the old location. Both distributions are left-skewed, the distribution for the data of the old location is severely affected by the presence of outliers.

I hope this helps!