Respuesta :
Answer:
a) The Mean Value is 3.118
b) The Average Deviation from the Mean is 0.0416
c) The true standard deviation is approximately 0.0492
d) The standard error (deviation) of the mean, SEM is approximately 0.021982
e) The percentage error is approximately 0.751%
7. Part a;
The data set 'b' is more accurate
The set 'a' is more precise
Part b;
The number of significant figures used in reporting measured values is determined by number of digits that are actually known based on the measuring methods, with the inclusion of one more digit which is an approximation
Step-by-step explanation:
a) The given values of pi are;
3.14, 3.11, 3.20, 3.06, and 3.08
The number of data points, n = 5
Therefore, we have;
The mean, [tex]\bar x[/tex] = (∑x)/n = (3.14 + 3.11 + 3.20 + 3.06 + 3.08)/5 = 3.118
b) The Average Deviation from the Mean, MAD, is given as follows;
[tex]\dfrac{1}{n} \times \sum \limits_{i = 1}^n \left |x_i - \bar x \right |[/tex]
Therefore, we get;
[tex]MAD =\dfrac{1}{5} \times (\left |3.14 - 3.118 \right | +\left |3.11 - 3.118 \right | + \left |3.20 - 3.118 \right | + \left |3.06 - 3.118 \right | + \left |3.08 - 3.118 \right |) =0.0416[/tex]
The Average Deviation from the Mean = 0.0416
c) The true standard deviation, σ, is given as follows;
[tex]s =\sqrt{\dfrac{\sum \left (x_i-\bar x \right )^{2} }{n}}[/tex]
From MS Excel, we have;
[tex]\sum \left (x_i-\bar x \right )^{2}[/tex]= 0.01208
∴ s = √((0.01208)/((5)) ≈ 0.0491528229 ≈ 0.0492
d) The standard error of the mean, SEM is given as follows;
SEM = s/√n
∴ SEM = 0.0491528229/√5 ≈ 0.02198181065 ≈ 0.021982
e) The percentage error = (The correct value - The mean value)/(The correct value) × 100
The given correct value = 3.14159
∴ The percentage error = ((3.14159 - 3.118)/3.14159) × 100 ≈ 0.751%
7. Part a;
The given Book value = 7.86 g/cm³
A precise data are data that are close to each other, while an accurate data has value that is close to the true value
The mean of the data set are;
Set a. μₐ = (7.72 + 7.74 + 7.73 + 7.75 + 7.74)/5 = 7.736
The difference from the true value = 7.86 - 7.736 = 0.124
Set b. [tex]\mu_b[/tex] = (7.86 + 7.90 + 7.78 + 7.93 + 7.838)/5 = 7.8616
The difference from the true value = 7.86 - 7.8616 = -0.0016
Therefore, the data set 'b' is more accurate
The precision is given as follows;
The average deviation is found using the following formula;
[tex]\dfrac{1}{n} \times \sum \limits_{i = 1}^n \left |x_i - \mu \right |[/tex]
The average deviation of set a = ((7.736 - 7.72) + (7.74 - 7.736) + (7.75 - 7.736) +(7.736 - 7.73) + (7.74 - 7.736))/5 = 0.0088
The precision of set a = 7.736 ± 0.008
The average deviation of set b = ((7.8616 - 7.86) + (7.90 - 7.8616) + (7.8616 - 7.78) + (7.93 - 7.8616) + (7.8616 - 7.838))/5 = 0.04272
The precision of set b = 7.8616 ± 0.04272
due to the closeness (smaller average deviation) of set 'a' than 'b', the set 'a' is more precise
Part b;
What determines the number of significant figures used in reporting measured values is determined by the number of significant figures that is known with confidence and an extra approximation or estimated last digit.
