# Constructing Confidence Intervals

• What is a confidence interval?
• What are some factors that affect the size of a confidence interval?
• In the discussion for week 4, you rolled a pair of dice 10 times and calculated the average sum of your rolls. Then you did the same thing with 20 rolls. Use your results from the week 4 discussion for the average of 10 rolls and for the average of 20 rolls to construct a 95% confidence interval for the true mean of the sum of a pair of dice (assume σ = 2.41).
• What do you notice about the length of the interval for the mean of 10 rolls versus the mean of 20 rolls? Did you expect this? Why or why not?
• Using your mean for 20 rolls, calculate the 90% confidence interval. Explain its size as compared to the 95% confidence interval for 20 rolls.

This is my work from week 4:

Sampling a pair of die discussion question.

One die has six possible outcomes when it is rolled; while two die has 6^2 = 36 possible outcomes when rolled together. Rolling the die ten times produces a 6^10 possible outcomes. Rolling it 20 times produces a 6^20 possible outcomes.

Outcomes for the first ten rolls of a die

 First, die Second, die sum 4 6 10 3 1 4 5 2 7 3 5 8 5 4 9 5 2 7 1 2 3 6 1 7 2 5 7 2 4 6 Total = 68

Average = sum/number of rolls = 68/10 = 6.8

Outcomes for the second 20 rolls of a die

 First, die Second, die sum 3 2 5 4 6 10 5 6 11 5 6 11 2 2 4 1 3 4 5 5 10 2 6 8 1 4 5 5 5 10 3 6 9 1 2 3 2 2 4 6 5 11 3 5 8 2 2 4 3 3 6 2 5 7 6 4 10 1 1 2 Total = 142

Average = sum/number of rolls = 142/20 = 7.1

A central limit theorem is a theorem which allows us to work with approximately normal distribution by simplifying problems in statistics (Rosenblatt, 1956).

The central limit theorem states that the average of a sample distribution approaches normal distribution average as the size of the sample increases. It says that in a test of rolling a die, as the number of rolling a die increases the average of a distribution sample tends to be a normal distribution.

While testing the rolling of a pair of die, I’ve determined that the average of rolling a die is 7. These results show that when the number of rolling a die increases, the average leans towards seven thus proving the central limit theorem.