11. Let x? be the mean of a random sample of size 50 drawn...

11. Let x? be the mean of a random sample of size 50 drawn from a population with mean 112 and standard deviation 40. a) Find the probability that x? assumes a value between 110 and 114. b) Find the probability that x? assumes a value greater than 113.

Transcript

00:02 Now we have x to be equal to the mean of a random sample of size 50 drawn from a population with a population mean of 112 and a standard deviation of 40.

00:15 For the first question, you are to determine the probability that x actually assumes a value between 110 and 114.

00:33 So since we have a sample of 50 drawn out of the population, according to the central limit theorem, the mu of the sample is approximately equal to the mean of the population and that is equal to one hundred and twelve however the sample standard deviation is equal to the population standard deviation divided by the square width of the sample size so that will be equal to 40 divided by the square width of 50 so to find the probability we need to complete the z scores for the individual values so let's let's continue here so it becomes a probability that now the z score is equal to the x value minus the sample mean divided by the sample standard deviation so in this case we have 110 minus sample mean 112 divided by the sample standard that deviation that is this term here we have 40 divided by the square rate of 50 and this is less than or equal to z which is also less than or equal to 114 minus 112 divided by 40 over the square rate of 50 so we simplify both terms in the expression and we have the probability that negative 0 .3 5 4 is less than or equal to z, this less than or equal to 0 .354.

02:55 We're representing this on a standard normal curve, which has zero at the center.

03:04 You can assume this vertical line to be negative 0 .354 and this could be 0 .3 .4.

03:14 I'm interested in the region between these 2 z values and that will become the normal of 0 .354 minus the normal of negative 0 .3 5 4.

03:34 So this term is equal to 0 .6383 minus 0 .3617...

Question

11. Let x? be the mean of a random sample of size 50 drawn from a population with mean 112 and standard deviation 40. a) Find the probability that x? assumes a value between 110 and 114. b) Find the probability that x? assumes a value greater than 113.

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