Computing Confidence Interval Mean

Fictitious Stress Index (FSI) The Fictitious Stress Index (FSI) is a measure of a person's stress level. The FSI ranges from 0 to 70. A researcher acquires a random sample of undergraduates at Kennesaw State University, and administers the FSI to these students at the beginning of Fall Semester 2000. The FSI scores follow:   8 8 9 10 11 14 17 19 20 22 24 25 26 27 27 28 29 29 30 32 32 33 34 35 35 36 38 40 40 42 44 46 52 58 66

Mean Stress Level We wish to estimate the population mean FSI for undergraduates at Kennesaw State University.

Sample Statistics Item Value sample mean (m) 27.6923 sample size (n) 39 sample standard deviation (sd) 14.6631

Confidence Level Our next step is to select a confidence level this number will provide a level of confidence in our estimation process. A standard choice is 95% confidence. Using the table @ http://www.mindspring.com/~cjalverson/ztable.htm, we obtain the following row: 2.00 0.022750 0.95450 Our multiplier is 2.00.

Z(k) PROBRT PROBCENT 0.00 0.50000 0.00000 0.05 0.48006 0.03988 0.10 0.46017 0.07966 0.15 0.44038 0.11924 0.20 0.42074 0.15852 0.25 0.40129 0.19741 0.30 0.38209 0.23582 0.35 0.36317 0.27366 0.40 0.34458 0.31084 0.45 0.32636 0.34729 0.50 0.30854 0.38292 0.55 0.29116 0.41768 0.60 0.27425 0.45149 0.65 0.25785 0.48431 0.70 0.24196 0.51607 0.75 0.22663 0.54675 0.80 0.21186 0.57629 0.85 0.19766 0.60467 0.90 0.18406 0.63188 0.95 0.17106 0.65789 1.00 0.15866 0.68269 1.05 0.14686 0.70628 1.10 0.13567 0.72867 1.15 0.12507 0.74986 1.20 0.11507 0.76986 1.25 0.10565 0.78870 1.30 0.09680 0.80640 1.35 0.088508 0.82298 1.40 0.080757 0.83849 1.45 0.073529 0.85294 1.50 0.066807 0.86639 1.55 0.060571 0.87886 1.60 0.054799 0.89040 1.65 0.049471 0.90106 1.70 0.044565 0.91087 1.75 0.040059 0.91988 1.80 0.035930 0.92814 1.85 0.032157 0.93569 1.90 0.028717 0.94257 1.95 0.025588 0.94882 2.00 0.022750 0.95450 2.05 0.020182 0.95964 2.10 0.017864 0.96427 2.15 0.015778 0.96844 2.20 0.013903 0.97219 2.25 0.012224 0.97555 2.30 0.010724 0.97855 2.35 0.009387 0.98123 2.40 0.008198 0.98360 2.45 0.007143 0.98571 2.50 0.006210 0.98758 2.55 0.005386 0.98923 2.60 0.004661 0.99068 2.65 0.004025 0.99195 2.70 .0034670 0.99307 2.75 .0029798 0.99404 2.80 .0025551 0.99489 2.85 .0021860 0.99563 2.90 .0018658 0.99627 2.95 .0015889 0.99682 3.00 .0013499 0.99730  

Lower Confidence Bound m ≈ 27.6923 sd  14.6631 Z = 2 lower bound = m – Z*sd/(n) ≈ 27.6923 – 2*14.6631/(39) ≈ 22.9963

Upper Confidence Bound m ≈ 27.6923 sd  14.6631 Z = 2 upper bound = m + Z*(sd/n) ≈ 27.6923 + 2*(14.6631/39) ≈ 32.3882

Write the Interval We write the approximate interval as [22.9963,32.3882].

Confidence Estimation Schematic Compute lower = m – Z*(sd/n) upper = m + Z*(sd/n) Compute m sd Population  Obtain Sample Size = n

Interpretation ─ Population and Mean We seek the population mean Fictitious Stress Index for undergraduates at Kennesaw State University.

Interpretation ─ Family of Samples We obtain random samples of n=39 students per sample. Our Family of Samples consists of every possible random sample as described above.

Interpretation ─ Family of Intervals From each member of the Family of Samples we comupute the interval [m - 2(sd/n), m + 2(sd/n)]. Our Family of Intervals consists of every possible interval computed as above.

Interpretation ─ Confidence Approximately 95% of the members of the Family of Intervals cover , the population mean Fictitious Stress Index for undergraduates at Kennesaw State University. The remaining 5% or so fail. We view our single interval, [22.9968, 32.3882]., as being drawn at random from the Family of Intervals. If our interval is drawn from the 95% supermajority, then the population mean Fictitious Stress Index for undergraduates at Kennesaw State University is between 22.9968 and 32.3882 .