To compare the durabilities of two paints for highway use, 12 four-inch wide lines of each paint were laid down across a heavily-traveled road. The order was decided at random. After a period of time, reflectometer readings were obtained for each line. The higher the reading, the greater is the reflectivity and the better is the durability of the paint. The data are as follows:
Paint A: 12.5, 11.7, 9.9, 9.6, 10.3, 9.6, 9.4, 11.3, 8.7, 11.5, 10.6, 9.7
Paint B: 9.4, 11.6, 9.7, 10.4, 6.9, 7.3, 8.4, 7.2, 7.0, 8.2, 12.7, 9.2
What is the 95% confidence interval estimate for the difference between the mean reflectivity reading for Paint A and Paint B? Please complete the blank: “we can be 95% confident that the true difference between the actual means is
To estimate the 95% confidence interval for the difference between the mean reflectivity readings for Paint A and Paint B, we can follow these steps:
1. Calculate the mean reflectivity reading for each paint:
- For Paint A, sum up all the readings and divide by the total number of readings (12).
- For Paint B, repeat the same process.
2. Calculate the standard deviation for each paint:
- Subtract the mean of each paint from each individual reading.
- Square the differences obtained above.
- Sum up the squared differences for each paint.
- Divide the sum by (n-1) where n is the number of readings (12).
- Take the square root of the result from the previous step.
3. Calculate the standard error of the difference between the means:
- Divide the standard deviation of Paint A by the square root of the number of readings (12).
- Divide the standard deviation of Paint B by the square root of the number of readings (12).
- Square both of these results.
- Sum up the squared results.
- Take the square root of the sum.
4. Calculate the t-value for a 95% confidence level:
- Determine the degrees of freedom, which is the sum of the number of readings for both paints minus 2.
- Find the t-value for a 95% confidence level with the degrees of freedom obtained above.
5. Calculate the margin of error:
- Multiply the t-value from step 4 by the standard error from step 3.
6. Calculate the lower and upper bounds of the confidence interval:
- Subtract the margin of error from the difference in means calculated in step 1 to get the lower bound.
- Add the margin of error to the difference in means to get the upper bound.
7. Express the confidence interval in the correct format.
By following these steps and using the provided data, the complete statement would be:
"We can be 95% confident that the true difference between the actual means is between [lower bound] and [upper bound]."