Standard Scores Questions
1. Use both the Student ID and Distance to Work variables.
2. List the Student ID at TESU in ascending order of Distance to Work.
3. Calculate the z-scores associated with each student (use the sample standard deviation for this calculation).
4. Identify potential outliers and explain your reasoning.
| ID | Distance to Work (whole miles) | Z-Score |
| 18 | 0 | (1.07) |
| 9 | 0 | (1.07) |
| 2 | 0 | (1.07) |
| 12 | 0 | (1.07) |
| 1 | 0 | (1.07) |
| 21 | 2 | (0.92) |
| 3 | 5 | (0.69) |
| 20 | 6 | (0.62) |
| 11 | 7 | (0.54) |
| 17 | 8 | (0.47) |
| 15 | 10 | (0.31) |
| 4 | 10 | (0.31) |
| 23 | 12 | (0.16) |
| 22 | 15 | 0.06 |
| 19 | 15 | 0.06 |
| 5 | 15 | 0.06 |
| 10 | 20 | 0.44 |
| 14 | 25 | 0.82 |
| 6 | 30 | 1.19 |
| 7 | 32 | 1.34 |
| 8 | 36 | 1.65 |
| 16 | 38 | 1.80 |
| 13 | 40 | 1.95 |
Confidence Intervals
1. Take a sample of the first four data points for the variable Distance to Work (unsorted - use the original order in the dataset).
2. Determine the 95% and 99% confidence intervals using the same size of 4.
3. Describe and compare the two intervals.
Solution
The confidence interval for 4-points is[x ̅-t_(1-α/2,3)×s/√4,x ̅+t_(1-α/2,3)×s/√4]
Mean = 3.75
Std Dev = 4.787
t-value for 95% CI = 3.182
t-value for 99% CI = 5.841
| 95% Confidence Interval | ||
| Lower | (3.87) | |
| Upper | 11.37 | |
| 99% Confidence Interval | ||
| Lower | (10.23) | |
| Upper | 17.73 |
The 99% Confidence interval is wider than 95% Confidence interval which is expected. The CI also contains negative value,which is absurd, but this has been estimated assuming normal distribution as underlying distribution and hence, contains negative values.
Take a sample of the first seven data points for the variable Distance to Work (unsorted - use the original order in the dataset).
Determine the 95% confidence interval. Use the same mean and SD, but change the sample size to 20 and determine the 95% confidence interval.
Describe and compare the two intervals.
The confidence interval for 20-points is [x ̅-t_(1-α/2,19)×s/√20,x ̅+t_(1-α/2,19)×s/√20]
| Mean | 16.00 | |
| Std Dev | 14.75 | |
| N | 20.00 | |
| t-value | 2.09 | |
| 95% CI | ||
| Lower | 9.10 | |
| Upper | 22.90 |
This interval is a lot narrower than the last ones. This is because the sample size is large which reduces the standard error of the mean and the t-value for large degrees of freedom is close to normal and smaller than the t-value of smaller degrees of freedom.
Student Database
| ID | School Enrolled | Months Enrolled | Birthday Month | Distance to Work (whole miles) | Height (whole inches) | Foot Size (whole inches) | Hand Size (whole inches) | Sleep (minutes) | Homework (minutes) |
| 1 | Arts and Sciences | 12 | January | 0 | 60 | 8 | 5 | 360 | 30 |
| 2 | Applied Science and Technology | 6 | February | 0 | 62 | 7 | 6 | 400 | 45 |
| 3 | Business and Management | 8 | April | 5 | 66 | 10 | 7 | 420 | 60 |
| 4 | Nursing | 10 | June | 10 | 68 | 12 | 8 | 440 | 15 |
| 5 | Public Service | 48 | July | 15 | 68 | 14 | 8 | 540 | 75 |
| 6 | Arts and Sciences | 48 | June | 30 | 70 | 12 | 9 | 480 | 120 |
| 7 | Applied Science and Technology | 36 | October | 32 | 72 | 11 | 8 | 320 | 80 |
| 8 | Applied Science and Technology | 32 | November | 36 | 75 | 14 | 7 | 440 | 60 |
| 9 | Business and Management | 3 | May | 0 | 67 | 10 | 7 | 360 | 180 |
| 10 | Business and Management | 3 | January | 20 | 62 | 7 | 6 | 360 | 180 |
| 11 | Applied Science and Technology | 32 | January | 7 | 69 | 10 | 7 | 420 | 180 |
| 12 | Applied Science and Technology | 11 | April | 0 | 67 | 11 | 8 | 390 | 210 |
| 13 | Applied Science and Technology | 3 | March | 40 | 68 | 9 | 6 | 480 | 120 |
| 14 | Applied Science and Technology | 6 | December | 25 | 69 | 11 | 7 | 390 | 150 |
| 15 | Applied Science and Technology | 7 | December | 10 | 74 | 11 | 8 | 450 | 120 |
| 16 | Applied Science and Technology | 1 | January | 38 | 67 | 10 | 7 | 360 | 150 |
| 17 | Business and Management | 36 | December | 8 | 65 | 10 | 7 | 360 | 210 |
| 18 | Business and Management | 1 | June | 0 | 66 | 7 | 7 | 360 | 120 |
| 19 | Applied Science and Technology | 14 | July | 15 | 70 | 10 | 8 | 420 | 120 |
| 20 | Applied Science and Technology | 12 | January | 6 | 73 | 11 | 8 | 480 | 120 |
| 21 | Applied Science and Technology | 12 | December | 2 | 69 | 11 | 8 | 390 | 120 |
| 22 | Applied Science and Technology | 12 | April | 15 | 73 | 11 | 8 | 420 | 180 |
| 23 | Applied Science and Technology | 1 | August | 12 | 70 | 11 | 7 | 300 | 240 |