RESPOND TO THESE WITH COMMNETS, OPNIONS AND APPRECIATION

1)The z score for George Washington is 72-70/4 = .5, and this means that George Washington's height is half a standard deviation higher than the mean. This is easy to see in this problem because the standard deviation is 4, and half of four is 2, and Washington is 2 inches taller than the mean. The probability that a randomly selected individual will be as tall or taller than George is .3085, so a 30.85% chance, and I got this by typing in 70 as the mean and 4 as the standard deviation in the normal calculator, then typing in X is greater than or equal to 72.The mean of the sampling distribution of 50 is the same as the population mean, 70. The standard deviation of the sampling distribution is 4/sqrt50 = .5657. Using the normal calculator, the probability of X being greater than or equal to 72 is .0002, or .02% chance.The mean of the sampling distribution of 100 remains the same as the population mean, 70. The standard deviation of the sampling distribution is 4/sqrt100 = .4. Using the normal calculator, the probability of X being greater than or equal to 72 is .0000002, or .00002% chance. The more people you have, the greater the chances are that the randomly selected person will be closer to the mean of 70 instead of 72 or greater. So, the mean remains the same, while the standard deviation of the mean decreases as the sample size increases. It makes sense also because the sample size is the denominator.

2)George Washington was 6 ft, or 72 in, tall. To find his z-score I did the following calculation z=72704 and got a result of z = .5. According to the chart showing the probabilities for a normal distribution, the probability of a randomly selected male having a height less than or equal to 72 in. is .6915. Conversely, the probability of a randomly selected male having a height greater than or equal to George Washington's height of 72 in. is 1-.6915, or .3085. The sampling distribution of a statistic is a probability distribution for all possible values of the statistic computed from a sample of size n. So if we are going to take a sample of 50 male heights, we are going to find the height probability distribution for a particular set of 50 men using n=50 as opposed to looking at the distribution for the whole population. The mean for the sample distribution is still the same as it is for the population, so we would be using a mean of 70.

If we were to increase our sample size to n=100, then the probabilities in the distribution would decrease and the standard deviation would decrease as well.