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 σ) of the mean (denoted by µ)".
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:
• The probability of obtain values within one deviation from the mean is 0.68
• The probability of obtain values within two deviation's from the mean is 0.95
• 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%
