Respuesta :
Answer:
[tex] P(27.8 < X <30.5)=P(X>27.8)-P(X>30.5)=0.025-0.0015=0.0235[/tex]
Step-by-step explanation:
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 lifespans of tigers in a particular zoo.
From the problem we have the mean and the standard deviation for the random variable X. [tex]\mu=120 , \sigma=110[/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
We want to find this probability:
[tex] P(27.8 < X <30.5)[/tex]
And in order to calculate how many deviation we are above/below the mean we can use the z score given by:
[tex]z =\frac{x-\mu}{\sigma}[/tex]
And if we use this formula for the two values given we have:
[tex] z_1 = \frac{27.8-22.4}{2.7}=2[/tex]
[tex] z_1 = \frac{30.5-22.4}{2.7}=3[/tex]
So we have values between 2 and 3 deviations above the mean.
We can use the following probabilities
[tex]P(X<\mu -\sigma)=P(X <19.7)=0.16[/tex]
[tex]P(X>\mu +\sigma)=P(X >25.1)=0.16[/tex]
[tex]P(X<\mu -2*\sigma)=P(X<17)=0.025[/tex]
[tex]P(X>\mu +2*\sigma)=P(X>27.8)=0.025[/tex]
[tex]P(X<\mu -3*\sigma)=P(X<14.3)=0.0015[/tex]
[tex]P(X>\mu +3*\sigma)=P(X>30.5)=0.0015[/tex]
And we can find this probability on this way:
[tex] P(27.8 < X <30.5)=P(X>27.8)-P(X>30.5)=0.025-0.0015=0.0235[/tex]
