Respuesta :
Answer:
83.85% of 1-mile long roadways with potholes numbering between 57 and 89
Step-by-step explanation:
The Empirical Rule states that, for a normally distributed(bell-shaped) random variable:
68% of the measures are within 1 standard deviation of the mean.
95% of the measures are within 2 standard deviation of the mean.
99.7% of the measures are within 3 standard deviations of the mean.
In this problem, we have that:
Mean = 65
Standard deviation = 8
Using the empirical (68-95-99.7) rule, what is the approximate percentage of 1-mile long roadways with potholes numbering between 57 and 89?
It is important to remember that the normal distribution is symmetric, which means that 50% of the measures are below the mean and 50% are avobe.
57 = 65 - 8
So 57 is one standard deviation below the mean.
By the Empirical Rule, 68% of the 50% below the mean is within 1 standard deviation of the mean.
89 = 65 + 3*8
So 89 is three standard deviations above the mean.
By the Empirical Rule, 99.7% of the 50% above the mean is within 3 standard deviations of the mean.
0.68*0.5 + 0.997*0.5 = 0.8385
83.85% of 1-mile long roadways with potholes numbering between 57 and 89
Answer:
For this case we want to find this probability:
[tex] P(57 <X<89)[/tex]
And we can calculate the number of deviations from the mean for the limits using the z score formula given by:
[tex] z = \frac{57-65}{8}= -1[/tex]
[tex] z = \frac{89-65}{8}= 3[/tex]
We know that within one deviation from the mean we have 68% of the values and on each tail we have (100-68)/2 % = 16%. And within 3 deviations we have 99.7% of the values and on each tail we have (100-99.7)/2% = 0.15%.
And the percentage desired would be:
(100%-0.15%) - 16% = 83.85%
Step-by-step explanation:
Previous concepts
The empirical rule, also known as three-sigma rule or 68-95-99.7 rule, "is a statistical rule which states that for a normal distribution, almost all data falls within three standard deviations (denoted by ฯ) of the mean (denoted by ยต)".
Let X the random variable who representt the number of potholes
From the problem we have the mean and the standard deviation for the random variable X. [tex]E(X)=65, Sd(X)=8[/tex]
So we can assume [tex]\mu=65 , \sigma=8[/tex]
On this case in order to check if the random variable X follows a normal distribution we can use the empirical rule that states the following:
โข The probability of obtain values within one deviation from the mean is 0.68
โข The probability of obtain values within two deviation's from the mean is 0.95
โข The probability of obtain values within three deviation's from the mean is 0.997
Solution to the problem
For this case we want to find this probability:
[tex] P(57 <X<89)[/tex]
And we can calculate the number of deviations from the mean for the limits using the z score formula given by:
[tex] z = \frac{57-65}{8}= -1[/tex]
[tex] z = \frac{89-65}{8}= 3[/tex]
We know that within one deviation from the mean we have 68% of the values and on each tail we have (100-68)/2 % = 16%. And within 3 deviations we have 99.7% of the values and on each tail we have (100-99.7)/2% = 0.15%.
And the percentage desired would be:
(100%-0.15%) - 16% = 83.85%
