Current research indicates that the distribution of the life expectancies of a certain protozoan is normal with a mean of 48 days and a standard deviation of 10.5 days. Find the probability that a simple random sample of 36 protozoa will have a mean life expectancy of 51 or more days.

P ( x_bar ≥ 51 ) = 0.0432

Step-by-step explanation:

Solution:-

- The random variable "X" denotes:

                  X : life expectancies of a certain protozoan

- The variable "X" follows normal distribution.

                  X ~ Norm ( 48 , 10.5^2 )

- A sample of n = 36 days was taken.

- The sample is also modeled to be normally distributed:

                  x ~ Norm ( 48 , ( 10.5 / √n)^2 )

- The sample standard deviation s = 10.5 / √n = 10.5 / √36

                                                      s = 1.75

- We are to investigate the the probability of sample mean x_bar ≥ 51 days:

                 P ( x_bar ≥ 51 )

- Standardize the results, evaluate Z-score:

                 P ( Z ≥ ( x_bar - u ) / s ) =  P ( Z ≥ ( 51 - 48 ) / 1.75 )

                 P ( Z ≥ 1.7142 ).

- Use the standardized normal table and evaluate:

                 P ( Z ≥ 1.7142 ) = 0.0432                  

Hence,      P ( x_bar ≥ 51 ) = 0.0432

solution:

the given function is ,

      =

the three roots of this quadratic function is , 0 ,8 and 10.all the roots are real and rational.

if a quadratic equation is of the form ,

sum of the roots = a + b +c= 0+8+10=18=  

product of roots taken two at a time = a b + b c+ c a = 0+80+0=80 =  

product of roots = a b c = 0=  

there are 11 doors in a hall starting from 0 to 11.the entry in the hall is permitted through gate number 0 and exit is allowed through only gate number 8 and 10.so, three roots(0,8,10) of quadratic function is given by entry and exit route of this hall.