Respuesta :
Answer:
We want to approximate the following probability with a %:
[tex] P(41 < X< 65)[/tex]
And we can find the number of deviations from the mean for the limits with this formula called the z score:
[tex] z = \frac{41-65}{8}= -3[/tex]
[tex] z = \frac{65-65}{0}=0[/tex]
And using the empirical rule we know that within 3 deviation from the mean we have 0.997 of the area so then for the middle of the area we need to have 0.4985
So we can approximate the probability required as:
[tex]P(41 < X< 65) \approx 0.4985[/tex]
And the percent would be 49.85
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 represent the number of daily requests to replace lightbulbs.
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
We want to approximate the following probability with a %:
[tex] P(41 < X< 65)[/tex]
And we can find the number of deviations from the mean for the limits with this formula called the z score:
[tex] z = \frac{41-65}{8}= -3[/tex]
[tex] z = \frac{65-65}{0}=0[/tex]
And using the empirical rule we know that within 3 deviation from the mean we have 0.997 of the area so then for the middle of the area we need to have 0.4985
So we can approximate the probability required as:
[tex]P(41 < X< 65) \approx 0.4985[/tex]
And the percent would be 49.85
