Question 520:

Asked on November 23, 2008

Tags: Confidence Intervals , Median , t-statistic , mean , Standard Error of the Mean , Mode

1

Answer:

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The mean is calculated by adding up all values and dividing by the total.

1. Old: Sum is 356 divided by 10 =  a mean of 35.6
2. New: Sum is 268 divided by 10 = a mean of 26.8

The median is value at which half are above this point and half are below it. It is usually obtained by ordering the data from lowest to highest then picking the middle value. For even numbers we pick the mid-point or average between the two middle values.

1. Old: We pick the average of the middle values 35 and 38 getting us a median of 36.5
2. New: We pick the average of the middle values 27 and 28 getting us a median of 27.5.

The mode is the most frequently occurring value in a set. If all values are equally occurring, then there is no mode.

1. Old: All values appear only once so the mode is undefined.
2. New: This dataset actually has two modes at 26 and 31.

The variance and standard deviation are closely related. The standard deviation is the square root of the variance, so once we calculate it, we take the square root and get the standard deviation. To find the variance, we do the following:

1. Square the Sum of each values
   1. Old:  356^2 = 126736
   2. New: 268^2 = 71824
2. Divide the squared sum by the total number (10) (Avg Sum Squared)
   1. Old: 12673.6
   2. New: 7182.4
3. Square each individual value, then add up all these squared values (Sum of Squares)
   1. Old: 13080
   2. New: 7330
4. Subtract the Avg Sum Squared (from part 2) from the Sum of Squares from part 3
   1. Old:13080-12673.6 = 406.4
   2. New: 7330-7182.4 = 147.6
5. Divide this result by the total number minus 1 to get the sample variance
   1. Old: 406.4/9 = 45.156
   2. New: 147.6/9 = 16.4
6. The sample standard deviation is then the square root of the sample variance
   1. Old: 6.719
   2. New: 4.049

To calculate the 95% confidence intervals around the mean, we need to calculate the margin of error using the following steps:

1. Find the standard error of the mean (SEM) which is just the standard deviation divided by the square root of the sample size. We'll use the values from above.
   1. Old: 6.719/SQRT(10) = 2.214
   2. New: 4.049/SQRT(10) =  1.28
2. The margin of error is the SEM times a critical value from the t-distribution (since the sample size is small). To find the critical value we can look up the value in a t-table or use the excel function =TINV(.05,9) where the parameters are 1- confidence level (called alpha) and the degrees of freedom (sample size-1). It will be the same for both groups. We get the value 2.26. The margin of error for both groups are:
   1. Old: 2.214*2.26 = 4.807
   2. New: 1.28*2.26 = 2.89
3. The 95% confidence intervals are then the margin of error added and subtracted from the mean:
   1. Old: ( 30.79 to 40.40 )
   2. New: ( 23.9 to 29.69 )