Respuesta :
Answer:
[tex] \mu_p -\sigma_p = 0.74-0.0219=0.718[/tex]
[tex] \mu_p +\sigma_p = 0.74+0.0219=0.762[/tex]
68% of the rates are expected to be betwen 0.718 and 0.762
[tex] \mu_p -2*\sigma_p = 0.74-2*0.0219=0.696[/tex]
[tex] \mu_p +2*\sigma_p = 0.74+2*0.0219=0.784[/tex]
95% of the rates are expected to be betwen 0.696 and 0.784
[tex] \mu_p -3*\sigma_p = 0.74-3*0.0219=0.674[/tex]
[tex] \mu_p +3*\sigma_p = 0.74+3*0.0219=0.806[/tex]
99.7% of the rates are expected to be betwen 0.674 and 0.806
Step-by-step explanation:
Check for conditions
For this case in order to use the normal distribution for this case or the 68-95-99.7% rule we need to satisfy 3 conditions:
a) Independence : we assume that the random sample of 400 students each student is independent from the other.
b) 10% condition: We assume that the sample size on this case 400 is less than 10% of the real population size.
c) np= 400*0.74= 296>10
n(1-p) = 400*(1-0.74)=104>10
So then we have all the conditions satisfied.
Solution to the problem
For this case we know that the distribution for the population proportion is given by:
[tex] p \sim N(p, \sqrt{\frac{p(1-p)}{n}})[/tex]
So then:
[tex] \mu_p = 0.74[/tex]
[tex] \sigma_p =\sqrt{\frac{0.74(1-0.74)}{400}}=0.0219[/tex]
The empirical rule, also referred to as the 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). Broken down, the empirical rule shows that 68% falls within the first standard deviation (µ ± σ), 95% within the first two standard deviations (µ ± 2σ), and 99.7% within the first three standard deviations (µ ± 3σ).
[tex] \mu_p -\sigma_p = 0.74-0.0219=0.718[/tex]
[tex] \mu_p +\sigma_p = 0.74+0.0219=0.762[/tex]
68% of the rates are expected to be betwen 0.718 and 0.762
[tex] \mu_p -2*\sigma_p = 0.74-2*0.0219=0.696[/tex]
[tex] \mu_p +2*\sigma_p = 0.74+2*0.0219=0.784[/tex]
95% of the rates are expected to be betwen 0.696 and 0.784
[tex] \mu_p -3*\sigma_p = 0.74-3*0.0219=0.674[/tex]
[tex] \mu_p +3*\sigma_p = 0.74+3*0.0219=0.806[/tex]
99.7% of the rates are expected to be betwen 0.674 and 0.806
