00:01
In the question it is given that entrance test core of a certain university has a mean is equal to 74 and standard deviation sigma is equal to 6 .8.
00:12
Further it is given that test core follows the normal distribution.
00:16
Consider x be the test core and it follows the normal distribution with parameters 74 and 6 .8 square.
00:26
In the a part the question is if the student is selected at a random what is is the probability that his test score is less than 65.
00:35
That is probability that x is less than 65 which is equals to probability that x minus mu upon sigma is less than 65 minus mu upon sigma.
00:46
We know that x minus mu upon sigma is nothing but the z which is equal to probability that z is less than 65 minus 74 divided by sigma is 6 .8 which is equal to probability that z is less than minus 1 .32.
01:06
Observe minus 1 .32 in the standard normal table will get the value at 0 .0934.
01:15
Therefore the answer for a part is 0 .0934.
01:20
The question for b part is if the sample of 50 student is selected at a random what is the probability that mean test core of the group is greater than 75? we know that by the sampling distribution of the sample mean.
01:34
The sample mean is equal to population mean.
01:37
Therefore mu x bar is equal to mu is equal to 74 and standard deviation is equal to sigma upon square root of n which is equal to here we have to obtain the probability that x bar is greater than 75 which is equal to probability that x bar minus mu x bar divided by sigma x bar is greater than 75 minus new x bar divided by sigma x bar which is equals to probability that which is equal to probability that z is greater than 1 .04 which is nothing but 1 minus probability that z is less than 1 .04 observe 1 .04 in the standard normal table will get the value as 0 .0 8508 which is equals to 0 .1492...