Given the following statistical list:
13,17,19,22,25,27,29,29,32,and 33. Assuming a desired "Confidence Level" of 95%, find the confidence interval for the above scores.
19.8 < u(mean)< 29.4
14.6 < u < 33
13.4 < u < 22.7
11.1 < u < 18.5
I have 2 weeks to finish this class and I've been stuck on this for awhile. I really need some help. I might be paying for some tutoring even.
To find the confidence interval for the given statistical list, you need to follow these steps:
1. Calculate the mean (average) of the data set. To do this, add up all the values and divide by the total number of values. For the given numbers: 13 + 17 + 19 + 22 + 25 + 27 + 29 + 29 + 32 + 33 = 266. Divide 266 by 10 (the total number of values) to get the mean: 26.6.
2. Calculate the standard deviation of the data set. This measures the spread or dispersion of the data around the mean. To calculate this, you can either use a calculator or spreadsheet software, or manually follow the formula. Using the formula, you find the deviation of each value (the difference between the value and the mean), square the deviations, sum them up, divide by the number of values, and take the square root. For example, to calculate the standard deviation manually:
- Deviation of 13 = 13 - 26.6 = -13.6
- Squaring the deviation: (-13.6)^2 = 184.96
- Repeat for each value and sum up all the squared deviations: 184.96 + ... + (deviation for 33)^2 = X
- Divide X by the number of values (10 in this case), and take the square root: sqrt(X / 10) = Y
- The standard deviation is Y.
3. Determine the critical value based on the desired confidence level. For a confidence level of 95%, consult a standard normal distribution table or use a calculator to find the critical value. The critical value defines the range within which we expect the mean to fall. For a 95% confidence level, the critical value is approximately 1.96.
4. Calculate the margin of error. The margin of error is the range added/subtracted to the mean to form the confidence interval. It is calculated by multiplying the critical value (from step 3) by the standard deviation (from step 2), and then dividing by the square root of the number of values. Margin of error = (critical value * standard deviation) / sqrt(number of values).
5. Calculate the lower and upper bounds of the confidence interval. Subtract the margin of error (from step 4) from the mean to get the lower bound, and add it to the mean to get the upper bound.
Using these steps, let's calculate the confidence interval for the given data set:
Mean (u) = 26.6
Standard Deviation = (Calculate using the steps mentioned above)
Critical Value at 95% Confidence Level = 1.96 (from standard normal distribution table)
Number of Values = 10
Margin of Error = (1.96 * Standard Deviation) / sqrt(10)
Lower Bound = Mean - Margin of Error
Upper Bound = Mean + Margin of Error
Substitute the values to calculate the confidence interval:
Lower Bound = 26.6 - (1.96 * Standard Deviation) / sqrt(10)
Upper Bound = 26.6 + (1.96 * Standard Deviation) / sqrt(10)
Once you have calculated the standard deviation, follow the steps above to compute the confidence interval. Keep in mind that the exact values will depend on the specific calculation of the standard deviation.
I would suggest working through these steps carefully and double-checking your calculations. If you are still having trouble, considering seeking tutoring assistance can be helpful. A tutor can guide you through the process and answer any questions you may have.