95% confidence interval for the difference between the proportions of service contracts sold
on treadmills versus exercise bikes is calculated by finding the mean, standard deviation, and
margin of error of the service contracts for the treadmills and exercise bikes sold. The mean
is calculated by adding the two figures which are 67 and 55 to get a sum of 122, which is then
dived by 2 to get 61.
The standard deviation is calculated by finding the square root of the variance. The variance
is arrived at by finding the difference between each of the two numbers from the calculated
mean. This is (67 – 61) and (55 – 61) which are equal to 6 and -6 respectively. These
differences are individually squared and then summed up. This is 62 + -62 which is equal to
36 + 36 that gives you 72. The summation is divided by the number of figures summed which
is 2 to give you the variance of 36. The square root of this variance is calculated to find the
standard deviation which is 6.
The margin of error is arrived at by finding the critical value which you then multiply by the
standard error. The critical value is calculated by converting the 95% to a decimal figure of
0.95 and then dividing it by 2. The answer is 0.475. This figure is ran against the Z table to
get its corresponding value which is 1.96. the standard error is calculated by dividing the
standard deviation by the square root of the number of treadmills added to the exercise bikes.
This is 6 + 11.05 = 17.05. The next step is to find the margin of error by multiplying the
critical value by the standard error, that is, 1.96 by 0.95 which is equal to 1.86.
With the mean, standard deviation, and margin of error all available, the confidence interval
can be calculated. This is by adding the margin of error to the mean and subtracting the two
variables. That is, 61 + 1.86 = 62.86 and 61 – 1.86 = 60.86.
From the latter calculations, 95% confidence interval is between 60.86 and 62.86.
on treadmills versus exercise bikes is calculated by finding the mean, standard deviation, and
margin of error of the service contracts for the treadmills and exercise bikes sold. The mean
is calculated by adding the two figures which are 67 and 55 to get a sum of 122, which is then
dived by 2 to get 61.
The standard deviation is calculated by finding the square root of the variance. The variance
is arrived at by finding the difference between each of the two numbers from the calculated
mean. This is (67 – 61) and (55 – 61) which are equal to 6 and -6 respectively. These
differences are individually squared and then summed up. This is 62 + -62 which is equal to
36 + 36 that gives you 72. The summation is divided by the number of figures summed which
is 2 to give you the variance of 36. The square root of this variance is calculated to find the
standard deviation which is 6.
The margin of error is arrived at by finding the critical value which you then multiply by the
standard error. The critical value is calculated by converting the 95% to a decimal figure of
0.95 and then dividing it by 2. The answer is 0.475. This figure is ran against the Z table to
get its corresponding value which is 1.96. the standard error is calculated by dividing the
standard deviation by the square root of the number of treadmills added to the exercise bikes.
This is 6 + 11.05 = 17.05. The next step is to find the margin of error by multiplying the
critical value by the standard error, that is, 1.96 by 0.95 which is equal to 1.86.
With the mean, standard deviation, and margin of error all available, the confidence interval
can be calculated. This is by adding the margin of error to the mean and subtracting the two
variables. That is, 61 + 1.86 = 62.86 and 61 – 1.86 = 60.86.
From the latter calculations, 95% confidence interval is between 60.86 and 62.86.
