Find the 98% confidence interval for estimating ìd based on these paired data and assuming normality. (Give your answers correct to one decimal place.)
Before
68,58,64,62,58,67
After
40,42,46,37,45,38
Lower limit:
Upper limit:
You'll need to start by finding a confidence interval for 2 samples.
Here is one you might be able to use:
CI = (mean difference) ± t-value (standard deviation difference/√n)
You will need to calculate the means and standard deviations.
T-value is found using a t-table for 98% confidence.
n = sample size
Once you have the interval calculated, you will have the lower and upper limits.
To find the 98% confidence interval for estimating μd (the population mean difference), we can use the formula:
Confidence interval = X̄d ± tα/2 * (sd/√n)
Where:
- X̄d is the sample mean difference,
- tα/2 is the critical t-value for the desired confidence level,
- sd is the standard deviation of the sample differences,
- n is the number of paired observations.
Let's calculate each component of the formula:
First, let's calculate the sample mean difference, X̄d:
1. Subtract each "After" observation from its corresponding "Before" observation to get the differences: (68-40), (58-42), (64-46), (62-37), (58-45), (67-38).
2. Calculate the mean of these differences. Sum all the differences and divide by the number of differences, which is 6.
Next, let's calculate the standard deviation of the sample differences, sd:
1. Calculate the squared differences for each pair by subtracting each difference from the mean difference and squaring the result.
2. Calculate the mean of these squared differences (variance).
3. Take the square root of the variance to get the standard deviation.
To get the critical t-value, tα/2, we need the degrees of freedom (df). The df for a paired t-test is given by n - 1, where n is the number of paired observations.
Once we have all these values, we can calculate the confidence interval using the formula mentioned above.
Now, let's go step by step and calculate the values needed.
1. Calculate X̄d:
X̄d = (68-40 + 58-42 + 64-46 + 62-37 + 58-45 + 67-38) / 6
2. Calculate the squared differences:
(68-40 - X̄d)^2, (58-42 - X̄d)^2, (64-46 - X̄d)^2, (62-37 - X̄d)^2, (58-45 - X̄d)^2, (67-38 - X̄d)^2
3. Calculate the mean of squared differences (variance).
4. Take the square root of the variance to get the standard deviation, sd.
5. Calculate the critical t-value, tα/2, with a 98% confidence level and n - 1 degrees of freedom.
6. Calculate the confidence interval using the formula:
Confidence interval = X̄d ± tα/2 * (sd/√n)
Substituting the calculated values into the formula will give you the lower limit and upper limit of the 98% confidence interval for estimating μd.