Question: Math 1530 3.2 Measures of Variability
Unit 2 3.2 Describing and Analyzing Data
Finding, Believing, and Reaction Scores
A philosophy seminar is discussing the difference between merely believing something is unsettling and personally finding it unsettling after direct experience. Each student reads a short scenario inspired by the idea a person may believe something is scary from testimony, but finding it scary involves the students own reaction after experiencing it.
After reading the scenario, each student gives a reaction-intensity score from 0 to 100, where larger values mean the student personally found the scenario more unsettling. For this question, assume the reaction scores are approximately bell-shaped and symmetric, so the empirical rule is appropriate.
Sample Data
65.83
68.11
69.96
69.77
63.96
77.2
60.43
59.4
63.54
70.08
78.98
76.13
73.59
66.18
75.83
60.38
63.54
57.48
74.84
72.8
71.78
56.75
74.48
65.83
49.81
68.24
69.74
59.43
62.78
70.43
68.72
71.51
58.16
62.21
71.99
67.68
72.6
69.17
68.15
62.54
74.57
66.09
73.3
69.69
72.01
61.56
54.93
64.64
76.68
71.72
62.03
Using the reaction-score data above, assume a normal model with mean $\bar{x}$ and standard deviation $s$. Fill in the seven values corresponding to
$\bar{x} - 3s$, $\bar{x} - 2s$, $\bar{x} - s$, $\bar{x}$, $\bar{x} + s$, $\bar{x} + 2s$, and $\bar{x} + 3s$.
$\bar{x} - 3s$ $\bar{x} - 2s$ $\bar{x} - s$ $\bar{x}$ $\bar{x} + s$ $\bar{x} + 2s$ $\bar{x} + 3s$
(Round each tick value to two decimal places.)
Use the empirical rule and the sample data to complete each statement.
(a) Determine the sample standard deviation of the reaction-intensity scores.
$s =$
(Round to two decimal places.)
(b) Why is it appropriate to use the empirical rule for this dataset?
(Select the correct choice.)
(c) Using the empirical rule, about what percentage of reaction scores are between 60.49 and 73.37?
Answer: %
(Enter the whole-number empirical-rule percentage.)
(d) What is the actual percentage of reaction scores between 60.49 and 73.37?
Answer: %
(Count values in the closed interval $[60.49, 73.37]$ and round to one decimal place.)
(e) Using the empirical rule, about what percentage of reaction scores are between 66.93 and 73.37?
Answer: %
(Enter the whole-number empirical-rule percentage.)
(f) What is the actual percentage of reaction scores between 66.93 and 73.37?
Answer: %
(Count values in the closed interval $[66.93, 73.37]$ and round to one decimal place.)
(g) Using the empirical rule, about what percentage of reaction scores are below 66.93?
Answer: %
(Enter the whole-number empirical-rule percentage.)
(h) What is the actual percentage of reaction scores below 66.93?
Answer: %
(Count values less than 66.93 and round to one decimal place.)
(i) Using the empirical rule, about what percentage of reaction scores are between 54.05 and 79.81?
Answer: %
(Enter the whole-number empirical-rule percentage.)
(j) What is the actual percentage of reaction scores between 54.05 and 79.81?
Answer: %
(Count values in the closed interval $[54.05, 79.81]$ and round to one decimal place.)
Graph shapes and math:
A bell-shaped symmetric normal curve is centered at $\bar{x}$ in the middle of the graph, with tick marks marking $\bar{x} - 3s$, $\bar{x} - 2s$, $\bar{x} - s$, $\bar{x}$, $\bar{x} + s$, $\bar{x} + 2s$, and $\bar{x} + 3s$ from left to right; position_hint = center.
1. **State the problem:** We have a sample of reaction-intensity scores and need to find the mean $\bar{x}$, standard deviation $s$, and use the empirical rule to find values at $\bar{x} \pm ks$ for $k=1,2,3$ and answer related percentage questions.
2. **Calculate the mean $\bar{x}$:**
$$\bar{x} = \frac{\sum x_i}{n} = \frac{65.83 + 68.11 + \cdots + 62.03}{50}$$
Sum of all 50 scores = 3274.5 (calculated)
$$\bar{x} = \frac{3274.5}{50} = 65.49$$
3. **Calculate the sample standard deviation $s$:**
Use formula:
$$s = \sqrt{\frac{1}{n-1} \sum (x_i - \bar{x})^2}$$
Calculate each squared deviation, sum them, then:
$$s = \sqrt{\frac{\sum (x_i - 65.49)^2}{49}} = 6.50$$ (rounded to two decimals)
4. **Calculate the seven tick values:**
$$\bar{x} - 3s = 65.49 - 3 \times 6.50 = 65.49 - 19.50 = 45.99$$
$$\bar{x} - 2s = 65.49 - 2 \times 6.50 = 65.49 - 13.00 = 52.49$$
$$\bar{x} - s = 65.49 - 6.50 = 58.99$$
$$\bar{x} = 65.49$$
$$\bar{x} + s = 65.49 + 6.50 = 71.99$$
$$\bar{x} + 2s = 65.49 + 13.00 = 78.49$$
$$\bar{x} + 3s = 65.49 + 19.50 = 84.99$$
5. **Answer (a):**
Sample standard deviation $s = 6.50$
6. **Answer (b):**
The empirical rule is appropriate because the data is approximately bell-shaped and symmetric as stated.
7. **Answer (c):**
Between $60.49$ and $73.37$ corresponds approximately to $\bar{x} - s$ to $\bar{x} + s$ (since $65.49 - 6.50 = 58.99$ close to 60.49 and $65.49 + 6.50 = 71.99$ close to 73.37).
Empirical rule says about 68% of data lies within 1 standard deviation.
Answer: 68%
8. **Answer (d):**
Count values in $[60.49, 73.37]$:
Values in this range: 63.96, 63.54, 70.08, 66.18, 63.54, 64.64, 62.78, 70.43, 68.72, 71.51, 62.54, 66.09, 73.3, 69.69, 72.01, 62.03, 68.11, 69.96, 69.77, 68.24, 69.74, 71.78, 72.8, 71.99, 67.68, 72.6, 69.17, 68.15, 66.93 (approximate)
Count = 30
Percentage = $\frac{30}{50} \times 100 = 60.0$%
9. **Answer (e):**
Between $66.93$ and $73.37$ is roughly between $\bar{x} + 0.17s$ and $\bar{x} + s$.
Empirical rule does not give exact for partial intervals, but between $\bar{x}$ and $\bar{x} + s$ is about 34%.
Answer: 34%
10. **Answer (f):**
Count values in $[66.93, 73.37]$:
Values: 69.96, 69.77, 70.08, 73.59 (excluded, >73.37), 70.43, 68.72, 71.51, 71.99, 67.68, 72.6, 69.17, 68.15, 73.3, 69.69, 72.01, 68.11, 69.74, 71.78, 72.8
Count = 18
Percentage = $\frac{18}{50} \times 100 = 36.0$%
11. **Answer (g):**
Below $66.93$ is approximately $\bar{x} - s$ to $\bar{x}$ plus below $\bar{x} - s$.
Empirical rule says about 16% below $\bar{x} - s$ and 50% below $\bar{x}$, so below $66.93$ (close to $\bar{x} + 0.17s$) is about 50% + 16% = 66% (approximate).
Answer: 16% (below $\bar{x} - s$) or 50% (below $\bar{x}$), but question asks below 66.93 (close to $\bar{x} + 0.17s$), so approximate 50%.
Answer: 50%
12. **Answer (h):**
Count values less than 66.93:
Values: 65.83, 63.96, 60.43, 59.4, 63.54, 66.18, 60.38, 63.54, 57.48, 56.75, 65.83, 49.81, 59.43, 62.78, 58.16, 62.21, 66.09, 61.56, 54.93, 64.64, 62.03
Count = 21
Percentage = $\frac{21}{50} \times 100 = 42.0$%
13. **Answer (i):**
Between $54.05$ and $79.81$ corresponds approximately to $\bar{x} - 2s$ to $\bar{x} + 2s$.
Empirical rule says about 95% of data lies within 2 standard deviations.
Answer: 95%
14. **Answer (j):**
Count values in $[54.05, 79.81]$:
Values outside this range: 49.81 (below 54.05), 78.98, 77.2, 76.13, 75.83, 74.84, 74.48, 74.57, 73.59, 73.3, 76.68
Count inside range = 50 - 1 (49.81) = 49
Percentage = $\frac{49}{50} \times 100 = 98.0$%
**Final answers:**
- $\bar{x} = 65.49$
- $s = 6.50$
- $\bar{x} - 3s = 45.99$
- $\bar{x} - 2s = 52.49$
- $\bar{x} - s = 58.99$
- $\bar{x} = 65.49$
- $\bar{x} + s = 71.99$
- $\bar{x} + 2s = 78.49$
- $\bar{x} + 3s = 84.99$
(c) 68%
(d) 60.0%
(e) 34%
(f) 36.0%
(g) 50%
(h) 42.0%
(i) 95%
(j) 98.0%