Confidence Interval Equation with Known Standard Deviation
Assuming we use the known formula (Sect. 8.1) since we are given (o-) so we would not be using the T distribution table of the unknown standard deviation (Sect. 8.2). If this assumption is wrong everything below is drastically incorrect.
Confidence Interval = Sample Mean [x-} + the Z-value providing an area of a/2 in the upper tail of the standard normal probability distribution (times) the Standard Deviation (divided by) the Square root of the Sample size
1. Dice Rolled 10 Times
Sample Mean (Average) = 7.6
Know Standard Deviation = 2.41
Sample Size = 10
1. Confidence level (or confidence coefficient) I am going to use is 95% (.95). This gives a Z value providing an area of a/2 in the upper tail of a normal distribution of 1.960. (NOTE: I’m honestly at a loss as to WTH all that means but Table 8.1 on page 340 gives these values for 90%, 95% and 99% confidence levels.)
The formula now looks like sample mean (x-) + 1.96 (standard deviation / square root of sample size)
x- + 1.96 (2.41 / sq root of 10)
x- + 2.41 / 3.162
x- + .7621
The lower limit is
x- (-) .7621 = 7.6 – .7621 = 6.8379
The upper limit is
x- + .7621 = 7.6 + .7621 = 8.3621
A 95% confidence that the true mean of the sum of rolls of dice (based on a sample of 10 rolls) lies between the interval of 6.8379 and 8.3621
2. Dice rolled 20 times
Sample Mean (Average) = 6.15
Known Standard Deviation = 2.41
Sample Size = 20
Confidence level = 95%
(x-) + 1.96 (2.41 / sq root of 20)
x- + 1.96 (2.41 / 4.4721)
x- + 1.96 (.5388)
x- + 1.056
A 95% confidence that the true mean of the sum of rolls of dice (based on a sample of 20 rolls) lies between the interval of 5.094 and 7.206.
My intervals are opposite of what should be expected. The confidence intervals for the smaller sample size should be larger, and the confidence interval for the larger sample size should be smaller. My confidence intervals are the opposite of this. I noticed last week that my results were different than everyone else’s as the mean of the sum of 20 rolls was further away from the expected mean than the mean of 10 rolls. Given this week’s lesson I can only assume that my 20 roll sample is the “one off” not accounted for (5%) in the .95 confidence coefficient.
After rolling the dice 10 times per the instructions, the numbers were 3, 4, 9, 4, 10, 10, 8, 11, 8 and 5 with a total sum of 72 with an average of 7.2.
After rolling the dice 20 times per the instructions, the numbers were 7, 7, 7, 3, 6, 8, 8, 12, 8, 8, 5, 9, 4, 6, 6, 6, 11, 5, 6 and 7 with a total sum of 139 with an average of 6.95.
The confidence interval for the true mean is 1.819943 for the 10 times we roll the dice. The confidence interval for the true mean is 0.937543 for the 20 times we roll the dice.
I noticed that the length of the interval for the mean decreased from 10 to 20 which is odd because I thought it would be the complete opposite since you will have more numbers to work on.