Finance

5. State the main points of the Central Limit Theorem for a mean.

The CLT is the basis for many control charts because it allows us to take a non-normal distribution and normalize it by increasing the sample size. The Central Limit Theorem states that the mean of a sample is equal to the mean of the population, and that the variance in the sample is equal to the variance in the population divided by the sample size. Additionally, the distribution of the average usually normalizes over time as we increase the sample size, even if the parent distribution as a whole is not normal. The central limit theorem shows us that regardless of the shape of the original distribution, the average of the sampled distribution will approach normal. The CLT is important in research because it justifies and helps us understand the importance of sampling size and analysis in research.

8.46 A random sample of 10 miniature Tootsie Rolls was taken from a bag. Each piece was weighed on a very accurate scale. The results in grams were

3.087 3.131 3.241 3.241 3.270 3.353 3.400 3.411 3.437 3.477

(a) Construct a 90 percent confidence interval for the true mean weight.

Mean = 3.3048

Standard Deviation= 0.13199

T value= 1.8311

3.3048-1.8331*0.13199/sqrt(10)

3.3048+1.8331*0.13199/sqrt(10)

3.2283 to 3.3813

(b) What sample size would be necessary to estimate the true weight with an error of ± 0.03 grams with 90 percent confidence?

n= (z*s/E)^2

n= (1.645*0.1399/0.03)^2= 52.4

(c) Discuss the factors which might cause variation in the weight of Tootsie Rolls during manufacture.

Some factors which might cause variation in the weight of tootsie rolls during the manufacturing process are: calibration of machine, changes in ingredients, changes in humidity and moisture, variation in the thickness and size of the wrapping, variation in the amount of ink used to print the logo on the wrapper.