00:01
In the a part, we have to obtain the probability that single battery randomly selected from the population will have a life between 75 and 85.
00:09
Consider x be the life and we have to obtain the probability that 75 is less than or equal to x is less than or equal to 85.
00:17
In this case, the value of mean is given as 80 and standard deviation is equal to 10.
00:23
This probability is obtained by using the formula of inequality that is probability that x is less than 85 minus probability that x is less than 75 we know that z score is the standard normal variable with mean zero and standard deviation 1 and it is obtained by the formula x minus mu upon sigma is less than 8 95 minus mu upon sigma minus probability that x minus mu upon sigma is less than 75 minus mu upon sigma which is equals to probability that this is nothing but z is less than 85 minus 80 divided by 10 minus probability that z is less than 75 minus 80 divided by 10 which is equal to probability that z is less than 0 .5 minus probability that z is less than minus 0 .5 observe 0 .5 and minus 0 .5 in the standard normal table will get the values as 0 .61915 minus 0 .3085, which is equals to 0 .3830.
01:40
Therefore, the probability that a single battery randomly selected from the population will have a life between 75 and 85 is equal to 0 .3830.
01:52
In the v part, we have to obtain the probability that 9 randomly sample batteries from the population will have a sample mean life between 75 and 85.
02:05
That is probability that 75 is less than or equal to x bar less than or equal to 85.
02:12
We know that central limit theorem states that if the sample sizes increases that is greater than 30, then the sample mean follows normal distribution with mean equal to mean.
02:23
Mu x bar and standard deviation which is denoted as sigma x bar is equal to sigma upon square root of n here the value of mean equal to mu x bar which is equals to 80 and the value of standard is equal to sigma x bar is equal to sigma x bar is equal to sigma the probability that 75 is less than or equal to x bar is less than or equal to 85 is equal to probability that x bar is less than or equal to 85 minus probability that x bar is less than or equal to 75, which is equals to probability that x bar minus mu x bar divided by sigma x bar is less than or equal to 85 minus mu x bar divided by sigma x bar minus probability...