00:01
This problem says a sample of size 115 will be drawn from a population with a mean of 48 and a standard deviation of 12.
00:07
Find the probability that x will be between 45 and 50, and if this is correct that we want to find the probability that x is between 45 and 50, that means we're looking at a random selection from our population and we're not dealing with the sample mean of the sample size we were given, and to do that we would have to make our make an assumption that the population was normally distributed.
00:25
So we would use normal cdf in our calculator to get our probability, starting off with the lower bound and the upper bound that you want the probability between, which in this case would be 45 and 50, and then we follow that with the mean and standard deviation to finish our normal cdf, and the mean was given as 48 and the standard deviation is 12, and if this was the probability we were looking for, for the probability of a random selection for x, the probability rounded to four decimal places comes to 0 .1649.
00:54
If we were supposed to find the probability that our sample mean was between these two values, then that would change our notation and that would still allow us to use normal cdf to find our probability because our sample size is greater than or equal to 30, and that means the central limit theorem would apply if that was the case, and we were looking for the sample mean to be between these two values, and that means we can treat our sampling distribution as approximately normal, and we would still have the same lower bound and upper bound, but when we follow with our mean and standard deviation, it would be the mean of the sampling distribution and the standard deviation of the sample mean, and here the mean of our sampling distribution would be equivalent to the mean of the population, but the standard deviation the sample mean would be the standard deviation of the population divided by the square root of the sample size...