Original problem: Assume that a population is normally distributed with a mean of 100 and a standard deviation of 15. Would it be unusual for the mean of a sample of 3 to be 115 or more? Why or why not?
In this case the mean = 100 and a standard deviation = 15. The possibility of obtaining the mean of 115 for three samples that are randomly picked can be calculated as the probability of obtaining new mean minus the old mean divided by the standard deviation. The new standard error is equals to standard deviation divided by root of 3 where three is the number of the new sample: 15/1.732 = 8.66. This is the new standard deviation for the new sample. The probability that this will take place is:
P (X < 115) = P (Z < (115 – 100) /8.66) = P (X < 115) = P (Z< 15/8.66)
P (Z < 1.732) = 0.9582
This implies that the probability that the mean of the three samples picked randomly will be equivalent to 115 is 95.82%. This is a very big possibility. The main reason for this possibility is that in a normal distributed sample, the sample develops a normal curve in which there two extremes for the data. The average value therefore pulls the lower extremes and upper extremes to a common ground and thus, there is a high chance that a small sample selected randomly will record a higher mean than the current sample is quite high.
What if the size for each sample was increased to 20? Would a sample mean of 115 or more be considered unusual? Why or why not?
The addition of the sample to 20 may cause a decrease or an increase in the mean based on the actual position the added value lay in the normal curve of distribution. If they will be more at the center the mean will go up nut if there will be more at the extreme end the mean will go up. Therefore, it will not be unusual on obtaining the mean of 115 especially if more value are added at the center of the curve.
b Why is the Central Limit Theorem used?
The central limit theorem is of great importance since it implies that one can estimate certain statistics distribution even if one knows very little about the underlying distribution of the sample. Therefore it is commonly used since it eases the estimation process.
Consider situations in your work or home that could be addressed through a continuous probability distribution. Describe the situation and the variables, and determine whether the variables are normally distributed or not. How could you change these to a normally distributed dataset?
The situation that can be addressed using continuous probability in the workplace would be workers age. The company can set workers age to be between 20 and 45 years and thus formulating a continuous probability. In this case, the company will nullify anyone that is below 20 years and anyone above 45 years. There is also a chance that one value can have more than one individual such that there could be three or even more people sharing the same age. The variable here is age and employment. The variable may not be normally distributed since they are defined to concentrate within a certain range, in this case is between 20 and 45. To create normal distribution, the data can be grouped based on range and the frequency taken.