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# Statistics and Probability

Dataset 1
Assignment : Hypothesis Testing
Type the last three digits of your student number in the green cell:
18
XXXXXXXXXX
XXXXXXXXXX        320
Dataset 1
5
20        Groups            Frequencies
0.1        300    to    305    6     XXXXXXXXXX
0.05        305    to    310    10     XXXXXXXXXX
50    0.24    310    to    315    35     XXXXXXXXXX
0.05    0.03    315    to    320    81     XXXXXXXXXX
0    320    to    325    82     XXXXXXXXXX
0.19    325    to    330    38     XXXXXXXXXX
XXXXXXXXXX    0.4    330    to    335    14     XXXXXXXXXX
XXXXXXXXXX    0.55    335    to    340    10     XXXXXXXXXX
XXXXXXXXXX    0.78
XXXXXXXXXX    1
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Dataset 2
XXXXXXXXXX    Assignment : Hypothesis Testing
XXXXXXXXXX
XXXXXXXXXX    Dataset 2
20
0.1    Part (a)
195.41    206.38    202.46    192.86    199.80    199.28
0.05    193.39    204.68    201.22    193.10    200.95    197.85
193.09    203.03    202.54    195.15    203.07    196.72
194.93    205.17    200.41    194.62    205.25    198.16
196.98    207.46    198.35    195.17    205.79    198.81
198.43    205.31    198.27    195.28    205.74    201.03
200.59    203.61    196.06    197.41    203.93    200.54
Part (b)
202.75    201.64    195.85    199.48    201.73    202.58
204.12    203.89    195.08    201.49    201.47    201.61
202.64    196.85    200.48    202.73    203.58    205.12
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XXXXXXXXXX     XXXXXXXXXX     XXXXXXXXXX     XXXXXXXXXX     XXXXXXXXXX     XXXXXXXXXX
XXXXXXXXXX     XXXXXXXXXX     XXXXXXXXXX     XXXXXXXXXX     XXXXXXXXXX     XXXXXXXXXX
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Dataset 3
XXXXXXXXXX    Assignment : Hypothesis Testing
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XXXXXXXXXX    Dataset 3
20
0.1    0.75
1500    List (a)
10    1454.09    1563.76    1524.64    1428.59    1498.02    1492.81    1493.65
0.05    1433.88    1546.85    1512.24    1431.04    1509.46    1478.55    1482.82
1430.94    1530.29    1525.35    1451.53    1530.67    1467.24    1489.34
1449.28    1551.74    1504.14    1446.23    1552.51    1481.63
List (b)
1469.02    1573.89    1482.78    1450.94    1557.18    1487.38    1494.57
1483.50    1552.39    1481.97    1452.02    1556.66    1509.51    1506.01
1505.14    1535.33    1459.84    1473.31    1538.53    1504.68    1494.09
1526.73    1515.69    1457.80    1494.04    1516.54    1525.06    1505.98
1540.48    1538.11    1450.04    1514.20    1513.90    1515.39
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XXXXXXXXXX     XXXXXXXXXX     XXXXXXXXXX     XXXXXXXXXX     XXXXXXXXXX     XXXXXXXXXX
XXXXXXXXXX     XXXXXXXXXX     XXXXXXXXXX     XXXXXXXXXX     XXXXXXXXXX     XXXXXXXXXX
XXXXXXXXXX     XXXXXXXXXX     XXXXXXXXXX     XXXXXXXXXX     XXXXXXXXXX     XXXXXXXXXX
XXXXXXXXXX     XXXXXXXXXX     XXXXXXXXXX     XXXXXXXXXX     XXXXXXXXXX     XXXXXXXXXX
XXXXXXXXXX     XXXXXXXXXX     XXXXXXXXXX     XXXXXXXXXX     XXXXXXXXXX     XXXXXXXXXX
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Dataset 4
XXXXXXXXXX
XXXXXXXXXX    Assignment : Hypothesis Testing
20    Dataset 4
0.1
0.05        Resistance:
10
0.05        Motor running    Motor not running
0.17    10.34    11.14     XXXXXXXXXX    0.85
0.04    10.08    10.77     XXXXXXXXXX    0.74
XXXXXXXXXX    0.02    10.04    10.59     XXXXXXXXXX    0.6
XXXXXXXXXX    0.14    10.28    10.99     XXXXXXXXXX    0.76
XXXXXXXXXX    0.28    10.56    11.17     XXXXXXXXXX    0.66
XXXXXXXXXX    0.38    10.76    11.28     XXXXXXXXXX    0.57
XXXXXXXXXX    0.53    11.06    11.67     XXXXXXXXXX    0.66
XXXXXXXXXX    0.68    11.36    11.83     XXXXXXXXXX    0.52
XXXXXXXXXX    0.77    11.54    11.87     XXXXXXXXXX    0.38
XXXXXXXXXX    0.93    11.86    12.18     XXXXXXXXXX    0.37
XXXXXXXXXX    0.81    11.62    11.79     XXXXXXXXXX    0.22
XXXXXXXXXX    0.7    11.40    11.56     XXXXXXXXXX    0.21
XXXXXXXXXX    0.84    11.68    11.78     XXXXXXXXXX    0.15
XXXXXXXXXX    1    12.00    11.95     XXXXXXXXXX    0
XXXXXXXXXX    0.85    11.70    11.67     XXXXXXXXXX    0.02
XXXXXXXXXX    0.74    11.48    11.59     XXXXXXXXXX    0.16
XXXXXXXXXX    0.6    11.20    11.27     XXXXXXXXXX    0.12
XXXXXXXXXX    0.76    11.52    11.63     XXXXXXXXXX    0.16
XXXXXXXXXX    0.22    10.44    10.56     XXXXXXXXXX    0.17
XXXXXXXXXX    0.22    10.44    10.70     XXXXXXXXXX    0.31
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Dataset 5
XXXXXXXXXX
XXXXXXXXXX    Assignment : Hypothesis Testing
0.05
20.5    Dataset 5        102
0.95            0.7125
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XXXXXXXXXX    1    65    60.17     XXXXXXXXXX    0.75
0.95     XXXXXXXXXX    61.87    61.38     XXXXXXXXXX    0.58
50.95     XXXXXXXXXX    58.72    62.73     XXXXXXXXXX    0.43
XXXXXXXXXX     XXXXXXXXXX    56.52    62.55     XXXXXXXXXX    0.14
XXXXXXXXXX     XXXXXXXXXX    52.3    67.72     XXXXXXXXXX    0.5
XXXXXXXXXX     XXXXXXXXXX    56.43    66.52     XXXXXXXXXX    0.79
XXXXXXXXXX     XXXXXXXXXX    52.69    68.34     XXXXXXXXXX    0.65
XXXXXXXXXX    0    50    70.5     XXXXXXXXXX    0.69
XXXXXXXXXX     XXXXXXXXXX    52.43    67.15     XXXXXXXXXX    0.42
XXXXXXXXXX     XXXXXXXXXX    51.54    68.5     XXXXXXXXXX    0.54
XXXXXXXXXX     XXXXXXXXXX    54.77    64.46     XXXXXXXXXX    0.25
XXXXXXXXXX     XXXXXXXXXX    53.91    67.41     XXXXXXXXXX    0.64
XXXXXXXXXX     XXXXXXXXXX    56.21    64.03     XXXXXXXXXX    0.35
XXXXXXXXXX     XXXXXXXXXX    60.19    63.54     XXXXXXXXXX    0.74
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Dataset 6
Assignment : Hypothesis Testing
15    Dataset 6
0.12
G1    G2    G3
A    7    16    9
B    5    21    16
School    C    11    25    14
D    5    18    18
E    9    17    11
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XXXXXXXXXX    0.05     XXXXXXXXXX     XXXXXXXXXX
XXXXXXXXXX     XXXXXXXXXX     XXXXXXXXXX     XXXXXXXXXX
XXXXXXXXXX     XXXXXXXXXX     XXXXXXXXXX
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XXXXXXXXXX     XXXXXXXXXX     XXXXXXXXXX
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Reference
3
18
Student numbers    Last3        Seed value
B XXXXXXXXXX    204    0    0.05
B XXXXXXXXXX    224    0
B XXXXXXXXXX    476    0
B XXXXXXXXXX    935    0
B XXXXXXXXXX    479    0
B XXXXXXXXXX    662    0
B XXXXXXXXXX    463    0
B XXXXXXXXXX    837    0
B XXXXXXXXXX    309    0
B XXXXXXXXXX    219    0
B XXXXXXXXXX    85    0
B XXXXXXXXXX    353    0
B XXXXXXXXXX    414    0
B XXXXXXXXXX    800    0
B XXXXXXXXXX    347    0
B XXXXXXXXXX    307    0
B XXXXXXXXXX    44    0
B XXXXXXXXXX    110    0
B XXXXXXXXXX    967    0
B XXXXXXXXXX    464    0
B XXXXXXXXXX    570    0
B XXXXXXXXXX    650    0
B XXXXXXXXXX    882    0
B XXXXXXXXXX    304    0
B XXXXXXXXXX    276    0
B XXXXXXXXXX    488    0
B XXXXXXXXXX    585    0
B XXXXXXXXXX    295    0
B XXXXXXXXXX    300    0
B XXXXXXXXXX    346    0
B XXXXXXXXXX    534    0
B XXXXXXXXXX    448    0
B XXXXXXXXXX    233    0
B XXXXXXXXXX    32    0
B XXXXXXXXXX    458    0
B XXXXXXXXXX    56    0
B XXXXXXXXXX    582    0
B XXXXXXXXXX    439    0
B XXXXXXXXXX    196    0
B XXXXXXXXXX    627    0
B XXXXXXXXXX    455    0
B XXXXXXXXXX    814    0
B XXXXXXXXXX    322    0
B XXXXXXXXXX    901    0
B XXXXXXXXXX    724    0
B XXXXXXXXXX    328    0
B XXXXXXXXXX    853    0
B XXXXXXXXXX    7    1
B XXXXXXXXXX    463    0
B XXXXXXXXXX    522    0
B XXXXXXXXXX    878    0
B XXXXXXXXXX    983    0
B XXXXXXXXXX    503    0
B XXXXXXXXXX    367    0
B XXXXXXXXXX    975    0
B XXXXXXXXXX    51    0
B XXXXXXXXXX    146    0
B XXXXXXXXXX    765    0
B XXXXXXXXXX    210    0
B XXXXXXXXXX    959    0
B XXXXXXXXXX    834    0
B XXXXXXXXXX    572    0
B XXXXXXXXXX    67    0
B XXXXXXXXXX    640    0
B XXXXXXXXXX    863    0
B XXXXXXXXXX    876    0
B XXXXXXXXXX    39    0
B XXXXXXXXXX    956    0
B XXXXXXXXXX    73    0
B XXXXXXXXXX    969    0
B XXXXXXXXXX    10    1
B XXXXXXXXXX    688    0
B XXXXXXXXXX    187    0
B XXXXXXXXXX    882    0
B XXXXXXXXXX    112    0
B XXXXXXXXXX    282    0
B XXXXXXXXXX    654    0
B XXXXXXXXXX    373    0
B XXXXXXXXXX    176    0
B XXXXXXXXXX    24    0
B XXXXXXXXXX    972    0
B XXXXXXXXXX    339    0
B XXXXXXXXXX    312    0
B XXXXXXXXXX    10    1
B XXXXXXXXXX    95    0
B XXXXXXXXXX    610    0
B XXXXXXXXXX    198    0
B XXXXXXXXXX    430    0
B XXXXXXXXXX    754    0
B XXXXXXXXXX    841    0
B XXXXXXXXXX    311    0
B XXXXXXXXXX    946    0
B XXXXXXXXXX    852    0
B XXXXXXXXXX    676    0
B XXXXXXXXXX    770    0
B XXXXXXXXXX    18    0
B XXXXXXXXXX    255    0
B XXXXXXXXXX    502    0
B XXXXXXXXXX    838    0
B XXXXXXXXXX    111    0
B XXXXXXXXXX    30    0
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1. All submissions must be in the form of PDF documents. Spread-
sheets exported to PDF will be accepted, but calculations must
e annotated or explained.
2. It is up to you how you do the calculations in each question, but
you must explain how you a
calculation. This can be done with a written explanation and
y using the relevant equations, along with showing the results
of intermediate stages of the calculations. In other words, you
need to show that you know how to do a calculation for a statistic
3. Each one of the questions involves a statistical test. Marks within
each question will generally be awarded for:
1
All calculations should use this data
All calculations should use the datasets in the excel spreadsheet
Graphs should be done in excel too
â€¢ Deciding which statistical test to use,
â€¢ Framing your Hypotheses and proper conclusions,
â€¢ Identifying the parameters for the test and
â€¢ Showing a reasonable level of clarity, detail and explanation
in the calculations needed to ca
y out the test.
4. The data you have been given is in the worksheets of an Excel
to not try to circumvent this; if you wish to use a spreadsheet to
Question 1
The lifetimes (in units of 106 seconds) of certain satellite components
are shown in the frequency distribution given in â€˜Dataset1â€™.
1. Draw a frequency polygon, histogram and cumulative frequency
polygon for the data.
2. Calculate the frequency mean, the frequency standard deviation,
the median and the first and third quartiles for this grouped data.
3. Compare the median and the mean and state what this indicates
tion relates to your frequency polygon and histogram.
4. Explain the logic behind the equations for the mean and standard
deviation for grouped data, starting from the original equations
for a simple list of data values. (This does not just mean â€™explain
how the equations are usedâ€™.)
Page 2
5. Ca
y out an appropriate statistical test to determine whether the
data is normally distributed.
Question 2
A manufacturer of metal plates makes two claims concerning the
thickness of the plates they produce. They are stated here:
â€¢ Statement A: The mean is 200mm
â€¢ Statement B: The variance is 1.5mm2.
To investigate Statement A, the thickness of a sample of metal plates
produced in a given shift was measured. The values found are listed
in Part (a) of worksheet â€˜Dataset2â€™, with millimetres (mm) as unit.
1. Calculate the sample mean and sample standard deviation for the
data in Part (a) of â€™Dataset2â€™. Explain why we are using the phrase
â€™sampleâ€™ mean or sampleâ€™ standard deviation.
2. Set up the framework of an appropriate statistical test on State-
ment A. Explain how knowing the sample mean before ca
ying
out the test will influence the structure of your test.
3. Ca
y out the statistical test and state your conclusions.
To investigate the second claim, the thickness of a second sample of
metal sheets was measured. The values found are listed in Part (b) of
worksheet â€˜Dataset2â€™, with millimetres (mm) as unit.
1. Calculate the sample mean and then the sample variance and
standard deviation for the data in Part (b).
Page 3
2. Set up the framework of an appropriate statistical test on State-
ment B. Explain how knowing the sample variance before ca
y-
ing out the test would influence the structure of your test.
3. Ca
y out the statistical test and state your conclusions.
Question 3
A manager of an inter-county hurling team is concerned that his team
lose matches because they â€˜fade awayâ€™ in the last ten minutes. He
has measured GPS data showing how much ground particular players
cover within a given time period; this is the data in list (a) in worksheet
â€˜Dataset3â€™. He has acquired the co
esponding data from an opposing,
more successful team, which is given in list (b).
1. Calculate the sample mean and sample standard deviation for the
two sets of data.
2. Set up the frame work of an appropriate statistical test to deter-
mine whether there is a difference in the distances covered by the
two groups of players.
3. Explain how having the results of the calculations above in ad-
vance of doing your statistical test will influence the structure of
that test.
4. Ca
y out the statistical test and state your conclusions.
Question 4
A study was ca
ied out to determine whether the resistance of the
control circuits in a machine are lower when the machine motor is
Page 4
unning. To investigate this question, a set of the control circuits was
tested as follows. Their resistance was measured while the machine
motor was not running for a certain period of time and then again
while the motor was running. The values found are listed in worksheet
â€˜Dataset4â€™, with kilo-Ohms as the unit of measurement.
1. Set up the structure of an appropriate statistical test to determine
whether the resistance of the control circuit in a machine are
lower when the machine motor is running.
2. Explain how the order of subtraction chosen to calculate the dif-
ferences will influence the structure of the test.
3. Give a reason why the data is measured with the engine not run-
ning first and then with the engine running.
4. Explain how knowing the mean of the differences in advance will
influence the structure of your statistical test.
5. Ca
y out the statistical test and state your conclusions.
Question 5
A study was ca
ied out to determine the influence of a trace element
found in soil on the yield of potato plants grown in that soil, defined as
the weight of potatoes produced at the end of the season. A large field
was divided up into 14 smaller sections for this experiment. For each
section, the experimenter recorded the amount of the trace element
found (in milligrams per metre squared) and the co
esponding weight
of the potatoes produced (in kilograms). This information is presented
in the worksheet â€˜Dataset5â€™ in the Excel document. Define X as the
trace element amount and Y as the yield.
Page 5
1. Draw a scatterplot of your data set.
2. Calculate the coefficients of a linear equation to predict the yield
Y as a function of X.
3. Calculate the co
elation coefficient for the paired data values.
4. Set up the framework for an appropriate statistical test to estab-
lish if there is a co
elation between the amount of the trace ele-
ment and the yield. Explain how having the scatterplot refe
ed
to above and having the value of r in advance will influence the
5. Ca
y out and state the conclusion of your test on the co
elation.
6. Comment on how well the regression equation will perform based
on the results above.
Question 6
A multinational corporation is conducting a study to see how its em-
ployees in five different countries respond to three gifts in an incentive
scheme. The numbers of employees who choose each of the three gifts
(G1 to G3) in each of the five countries (A to E) are given in the table
in â€˜Dataset6â€™ in the Excel document.
1. Set up the structure of an appropriate statistical test to deter-
mine whether the data supports a link between choice of gift and
country, including the statistic to be used.
2. Ca
y out this test, showing clearly in your work how the expected
values are calculated for your test statistic.
Page 6
Answered 1 days After May 04, 2023

## Solution

Atul answered on May 06 2023
Question 1
Groups Frequencies
300 to 305 6
305 to 310 10
310 to 315 35
315 to 320 81
320 to 325 82
325 to 330 38
330 to 335 14
335 to 340 10
The lifetimes (in units of 106 seconds) of certain satellite components are shown in the
frequency distribution given in â€˜Dataset1â€™.
1. Draw a frequency polygon, histogram and cumulative frequency polygon for the
data.
To draw the frequency polygon, we first need to calculate the midpoints of each group:
Intervals Frequencies Midpoint Cumulative Frequency
300-305 6 302.5 6
305-310 10 307.5 16
310-315 35 312.5 51
315-320 81 317.5 132
320-325 82 322.5 214
325-330 38 327.5 252
330-335 14 332.5 266
335-340 10 337.5 276
Histogram
Finally, to draw the cumulative frequency polygon, we need to calculate the cumulative
frequencies:

Intervals Frequencies Midpoint
Cumulative
Frequency
300-305 6 302.5 6
305-310 10 307.5 16
310-315 35 312.5 51
315-320 81 317.5 132
320-325 82 322.5 214
325-330 38 327.5 252
330-335 14 332.5 266
335-340 10 337.5 276
To calculate the frequency mean, we need to first calculate the midpoint of each interval, then
multiply each midpoint by its co
esponding frequency, sum up the results, and finally divide
y the total frequency.
Intervals Frequencies Midpoint
300-305 6 302.5
305-310 10 307.5
310-315 35 312.5
315-320 81 317.5
320-325 82 322.5
325-330 38 327.5
330-335 14 332.5
335-340 10 337.5
Frequency Mean = (6*302.5 + 10*307.5 + 35*312.5 + 81*317.5 + 82*322.5 + 38*327.5 +
14*332.5 + 10*337.5) / (6+10+35+81+82+38+14+10) = 320.7
The frequency standard deviation can be calculated using the following formula:
Ïƒ = sqrt[(Î£(f(x) - mean)^2) / n]
where f(x) is the frequency of each interval, mean is the frequency mean we just calculated,
and n is the total frequency.
f(x) midpoint deviation (deviation)^2 f(x)*(deviation)^2
6 302.5 -18.2 331.24 1987.44
10 307.5 -13.2 174.24 1742.4
35 312.5 -8.2 67.24 2353.4
81 317.5 -2.2 4.84 392.04
82 322.5 2.8 7.84 642.88
38 327.5 7.8 60.84 2312.92
14 332.5 12.8 163.84 2293.76
10 337.5 17.8 316.84 3168.4
Ïƒ = sqrt[(Î£(f(x) - mean)^2) / n] = sqrt[ (1987.44 + 1742.4 + 2353.4 + 392.04 + 642.88 +
2312.92 + 2293.76 + 3168.4) / 336] â‰ˆ 8.05
To find the median, we need to find the interval that contains the 168th value (the halfway
point between the 336 frequencies). The cumulative frequency column tells us that the 168th
value falls within the 320-325 interval, which has a cumulative frequency of 132. The
interval width is 325-320 = 5, and we need to find how much of this interval contains the
168th value. To do so, we calculate:
p = (168 - 132) / 82 = 0.439
Median = lower limit of the interval + (p * interval width) = 320 + (0.439 * 5) = 322.195
quartile = lower limit of the interval + (p * interval width)
where p is the fractional part of (n * quartile number) / 4 and n is the total frequency.
For the first quartile (Q1), we need to find the interval that contains the 84th value (which is
(336 * 1) / 4). The cumulative frequency column tells us that the 84th value falls within the
310-315 interval, which has a cumulative frequency of 16 + 35 = 51. The interval width is
315-310 = 5, and we need to find how much of this interval contains the 84th value. To do so,
we calculate:
p = (84 - 51) / 81 = 0.407
Q1 = lower limit of the interval + (p * interval width) = 310 + (0.407 * 5) = 312.035
For the third quartile (Q3), we need to find the interval that contains the 252nd value (which
is (336 * 3) / 4). The cumulative frequency column tells us that the 252nd value falls within
the 325-330 interval, which has a cumulative frequency of 132 + 82 + 38 = 252. The interval
width is 330-325 = 5, and we need to find how much of this interval contains the 252nd
value. To do so, we calculate:
p = (252 - 132 - 82) / 38 = 0.842
Q3 = lower limit of the interval + (p * interval width) = 325 + (0.842 * 5) = 329.21
Therefore, the first quartile (Q1) is approximately 312.035 and the third quartile (Q3) is
approximately 329.21.
3. Compare the median and the mean and state what this indicates about the
distribution. Comment on how the answer to this question relates to your frequency
polygon and histogram.
The median for this grouped data is approximately 321.875, and the mean is approximately
322.195.
Since the mean and the median are relatively close in value, this suggests that the data is
fairly symmetrically distributed. This is also evident from the frequency polygon and
histogram, where we see that the distribution is somewhat bell-shaped, with the highest
frequencies occu
ing in the middle of the data range and decreasing as we move towards the
extremes.
However, there is a slight right skew in the distribution, as we can see from the frequency
polygon and histogram where the right tail extends further than the left tail. This skewness is
also reflected in the fact that the mean is slightly larger than the median, indicating that the
ight tail of the distribution is pulling the mean towards it.
Overall, we can conclude that the distribution is roughly symmetric but slightly skewed to the
ight.
4. Explain the logic behind the equations for the mean and standard deviation for
grouped data, starting from the original equations for a simple list of data values. (This
does not just mean â€™explain how the equations are usedâ€™.)
The equations for the mean and standard deviation for grouped data are modifications of the
equations for the mean and standard deviation for a simple list of data values. The main
difference is that the grouped data is divided into intervals, and the frequency of each interval
is used to determine the weight of each interval in the calculation of the mean and standard
deviation.
For the mean, the equation for grouped data is:
mean = Î£ (midpoint * frequency) / Î£ frequency
where midpoint is the midpoint of each interval, and frequency is the frequency of each
interval. The numerator represents the sum of the products of the midpoint and frequency of
each interval, while the denominator represents the total frequency of all intervals. This
equation is used to calculate the weighted average of the midpoints of the intervals, where the
weight of each interval is its frequency.
For the standard deviation, the equation for grouped data is:
standard deviation = sqrt(Î£ [(x - mean)^2 * frequency] / (Î£ frequency - 1))
where x is the midpoint of each interval, mean is the mean of the data set, and frequency is
the frequency of each interval. The numerator represents the sum of the products of the
squared differences between the midpoint and the mean and the frequency of each interval,
while the denominator represents the total frequency of all intervals minus one. This equation
is used to calculate the weighted average of the squared deviations of the midpoints from the
mean, where the weight of each interval is its frequency.
The modification of the equations is necessary because grouped data provides less
information about the individual data points than a simple list of values. The midpoint of each
interval is used to represent all the data points within the interval, and the frequency of each
interval is used to determine the weight of each interval in the calculation of the mean and
standard deviation.
5.Ca
y out an appropriate statistical test to determine whether the data is normally
distributed.
To test for normality, we can use the Shapiro-Wilk test, which is a commonly used statistical
test for normality.
However, since our data is grouped and we only have the frequencies for each interval, we
cannot...
SOLUTION.PDF