The performances of a group of interns are evaluated by T
The performances of a group of interns are evaluated by their supervisors at the end of their internships. Their scores are: 55, 47, 62, 27, 50, 49, 66, 53, 50, 44, 63, 59. Complete the calculations below using this data. Show all of your work and clearly label each of your calculations. the mean the median the range the standard deviation the variance
The Anxiety General Stress Test (ANGST) has been designed to gauge the level of psychological stress management trainees experience when they are under pressure. For a random sample of trainees, the scores are as follows: 47, 49, 53, 53, 54, 58, 61, 64, 75, 81. Complete the calculations below using this data. Show all of your work and clearly label each of your calculations. What is the z score equivalent of ANGST = 81? What is the probability that someone selected at random will score 81 or lower? What percentage of all trainees will score between 60 and 75? Copy and paste your calculations for the three problems from Excel to Word and submit one Word document with your clearly labeled calculations and interpretations.
Paper For Above instruction
### Introduction
Statistical analysis plays a crucial role in evaluating performance metrics in research and practical applications. This paper details the computation of key descriptive statistics—mean, median, range, variance, and standard deviation—for two data sets. The first data set encompasses scores of interns evaluated at their internship completion, while the second set pertains to scores from the Anxiety General Stress Test (ANGST). Moreover, the paper applies inferential statistics to determine Z-scores, probabilities, and percentages related to the ANGST scores, offering insights into stress levels among trainees. Accurate calculation and interpretation of these statistics provide valuable tools for assessing performance and psychological states in educational and clinical settings.
### Analysis of Intern Performance Scores
The first data set includes intern evaluation scores: 55, 47, 62, 27, 50, 49, 66, 53, 50, 44, 63, 59. To analyze these, the calculations for central tendency and variability are performed step by step.
**Mean Calculation:**
The mean score, a measure of central tendency, is computed by summing all scores and dividing by the
number of observations:
Sum of scores = 55 + 47 + 62 + 27 + 50 + 49 + 66 + 53 + 50 + 44 + 63 + 59 = 679
Number of scores = 12
Mean = 679 / 12 ≈ 56.58
**Median Calculation:**
Sorting the scores in ascending order: 27, 44, 47, 49, 50, 50, 53, 55, 59, 62, 63, 66
Since there are even numbers of observations, median is the average of the two middle scores:
Middle scores = 50 and 53
Median = (50 + 53) / 2 = 51.5
**Range Calculation:**
Range = Maximum score – Minimum score = 66 – 27 = 39
**Variance and Standard Deviation Calculation:**
Variance measures the average squared deviation from the mean, while the standard deviation is the square root of variance.
Calculating each deviation, squaring, summing, then dividing by n-1 (sample variance):
Deviations:
(55 - 56.58) ≈ -1.58 → 2.50
(47 - 56.58)
(62 - 56.58)
(27 - 56.58)
(50 - 56.58)
(49 - 56.58)
(66 - 56.58)
-9.58 → 91.83
5.42 → 29.39
-29.58 → 874.54
-6.58 → 43.29
-7.58 → 57.49
9.42 → 88.57
(53 - 56.58) ≈ -3.58 → 12.84
(50 - 56.58) ≈ -6.58 → 43.29
(44 - 56.58) ≈ -12.58 → 158.33
(63 - 56.58) ≈ 6.42 → 41.30
(59 - 56.58) ≈ 2.42 → 5.86
Sum of squared deviations ≈ 2.50 + 91.83 +
+ 158.33 + 41.30 + 5.86 ≈ 1358.23
Variance = 1358.23 / (12 - 1) ≈ 1358.23 / 11 ≈ 123.48
Standard deviation = √123.48 ≈ 11.11
### Analysis of ANGST Scores
The second data set involves scores: 47, 49, 53, 53, 54, 58, 61, 64, 75, 81. The calculations here focus on central tendency, variability, and probability statistics.
**Mean Calculation:**
Sum = 47 + 49 + 53 + 53 + 54 + 58 + 61 + 64 + 75 + 81 = 595
Number of scores = 10
Mean = 595 / 10 = 59.5
**Median Calculation:**
Scores in order: 47, 49, 53, 53, 54, 58, 61, 64, 75, 81
Median = (54 + 58) / 2 = 56
**Variance and Standard Deviation:**
Deviations from mean (59.5):
47 - 59.5 = -12.5 → 156.25
49 - 59.5 = -10.5 → 110.25
53 - 59.5 = -6.5 → 42.25
53 - 59.5 = -6.5 → 42.25
54 - 59.5 = -5.5 → 30.25
58 - 59.5 = -1.5 → 2.25
61 - 59.5 = 1.5 → 2.25
64 - 59.5 = 4.5 → 20.25
75 - 59.5 = 15.5 → 240.25
81 - 59.5 = 21.5 → 462.25
+ 462.25 ≈ 1054.25
Variance = 1054.25 / (10 - 1) ≈ 1054.25 / 9 ≈ 117.14
Standard deviation = √117.14 ≈ 10.82
### Inferential Statistics: Z-Score and Probability
The Z-score indicates how many standard deviations an element is from the mean. For an ANGST score of 81:
Z = (X - µ) / σ = (81 - 59.5) / 10.82 ≈ 21.5 / 10.82 ≈ 1.99
Using the standard normal distribution, P(Z ≤ 1.99) ≈ 0.9767 (97.67%)
**Percentage of trainees scoring between 60 and 75:**
Calculate Z-scores:
Z(60) = (60 - 59.5) / 10.82 ≈ 0.0463
Z(75) = (75 - 59.5) / 10.82 ≈ 1.44
Using standard normal tables:
P(Z ≤ 1.44) ≈ 0.9251
P(Z ≤ 0.0463) ≈ 0.5192
Percentage = [P(Z ≤ 1.44) - P(Z ≤ 0.0463)] * 100 ≈ (0.9251 - 0.5192) * 100 ≈ 40.59%
### Conclusions
This statistical analysis reveals that the interns' scores have a mean around 56.58 with significant spread, as indicated by the standard deviation. The ANGST scores display similar variability, with a mean of 59.5.
The Z-score of 1.99 for a score of 81 suggests it is near the upper end of the distribution, with approximately 97.67% of trainees scoring lower. About 40.59% of trainees score between 60 and 75, indicating a moderate range of stress levels among the sampled individuals. These calculations enable educators and psychologists to interpret performance and stress data meaningfully, supporting targeted interventions and assessments.
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