Question

Male Height vs Female Height
71.0 65.6
66.0 60.4
68.3 62.8
67.1 61.5
69.4 63.9
80.0 74.9
65.3 59.6
60.6 54.7
66.1 60.5
67.9 62.4
73.6 68.3
75.8 70.6
62.6 56.8
78.9 73.8
71.7 66.3
69.4 63.9
73.3 68.0
66.9 61.3
70.4 65.0
60.1 54.2
69.4 63.9
66.0 60.4
66.8 61.2
68.7 63.2
66.1 60.4
75.4 70.2
66.5 60.8
71.6 66.2
64.4 58.7
66.0 60.4
74.0 68.7
61.4 55.6
65.1 59.5
69.3 63.8
68.9 63.4
69.0 63.5
71.1 65.7
67.3 61.8
65.9 60.2
66.8 61.3
65.4 59.7

List the 5-number summary for each data set

Answer 1

Identify any outliers for each data set using the interquartile range method. Show your calculation.

Answer 2

c) Construct a comparative box plot for the pair of variables chosen above.
Describe the shape of each boxplot.

Answer 3

d) Explain whether Empirical Rule can be used to describe the chosen data sets in Part 3a. Justify your response with a thorough explanation. Hint: Do either have outliers and/or extreme skewness?

Answer 4

To compute the 5-number summary for each dataset (Male Height and Female Height), we need to follow these steps:

1. **Minimum**: The smallest number in the dataset.
2. **First Quartile (Q1)**: The median of the first half of the data.
3. **Median (Q2)**: The middle value of the data.
4. **Third Quartile (Q3)**: The median of the second half of the data.
5. **Maximum**: The largest number in the dataset.

Knowing that there are 40 data points in total, we will first sort both datasets.

### Male Height Data
Given values:
```
71.0, 66.0, 68.3, 67.1, 69.4, 80.0, 65.3, 60.6, 66.1, 67.9, 73.6, 75.8,
62.6, 78.9, 71.7, 69.4, 73.3, 66.9, 70.4, 60.1, 69.4, 66.0, 66.8, 68.7,
66.1, 75.4, 66.5, 71.6, 64.4, 66.0, 74.0, 61.4, 65.1, 69.3, 68.9, 69.0,
71.1, 67.3, 65.9, 66.8, 65.4
```

#### Sorting Male Heights:
```
60.1, 60.6, 61.4, 62.6, 64.4, 65.1, 65.3, 65.4, 65.9, 66.0, 66.0, 66.0,
66.1, 66.1, 66.5, 66.8, 66.8, 66.9, 67.1, 67.3, 67.9, 68.3, 68.7, 68.9,
69.0, 69.3, 69.4, 69.4, 69.4, 70.4, 71.0, 71.6, 71.7, 73.3, 73.6, 74.0,
75.4, 75.8, 78.9, 80.0
```

#### 5-Number Summary (Male Height)
- **Minimum:** 60.1
- **Q1:** (66.0 + 66.0) / 2 = 66.0
- **Median:** (69.4 + 69.4) / 2 = 69.4
- **Q3:** (71.7 + 73.3) / 2 = 72.5
- **Maximum:** 80.0

### Female Height Data
Given values:
```
65.6, 60.4, 62.8, 61.5, 63.9, 74.9, 59.6, 54.7, 60.5, 62.4, 68.3, 70.6,
56.8, 73.8, 66.3, 63.9, 68.0, 61.3, 65.0, 54.2, 63.9, 60.4, 61.2, 63.2,
60.4, 70.2, 60.8, 66.2, 58.7, 60.4, 68.7, 55.6, 59.5, 63.8, 63.4, 63.5,
65.7, 61.8, 60.2, 61.3, 59.7
```

#### Sorting Female Heights:
```
54.2, 54.7, 55.6, 56.8, 58.7, 59.5, 59.6, 59.7, 60.2, 60.4, 60.4, 60.4,
60.5, 60.8, 61.2, 61.3, 61.3, 61.5, 61.8, 62.4, 62.8, 63.2, 63.4, 63.5,
63.8, 63.9, 63.9, 64.0, 66.3, 68.0, 68.3, 70.2, 70.6, 74.9, 73.8, 68.7, 75.6
```

#### 5-Number Summary (Female Height)
- **Minimum:** 54.2
- **Q1:** (60.4 + 60.4) / 2 = 60.4
- **Median:** (61.5 + 61.8) / 2 = 61.65
- **Q3:** (63.9 + 64.0) / 2 = 63.9
- **Maximum:** 75.6

### Summary
- **Male Height Summary:**
- Minimum: 60.1
- Q1: 66.0
- Median: 69.4
- Q3: 72.5
- Maximum: 80.0

- **Female Height Summary:**
- Minimum: 54.2
- Q1: 60.4
- Median: 61.65
- Q3: 63.9
- Maximum: 75.6

Answer 5

To identify outliers using the interquartile range (IQR) method, we first need to compute the IQR itself. The IQR is defined as the difference between the third quartile (Q3) and the first quartile (Q1):

\[
\text{IQR} = Q3 - Q1
\]

Outliers are defined as any data points that fall below \( Q1 - 1.5 \cdot \text{IQR} \) or above \( Q3 + 1.5 \cdot \text{IQR} \).

### Male Height Outlier Calculation

#### From previous data:
- **Q1 (Male Height)** = 66.0
- **Q3 (Male Height)** = 72.5

#### Calculate IQR:
\[
\text{IQR} = Q3 - Q1 = 72.5 - 66.0 = 6.5
\]

#### Determine lower and upper bounds:
- **Lower Bound**:
\[
Q1 - 1.5 \cdot \text{IQR} = 66.0 - 1.5 \cdot 6.5 = 66.0 - 9.75 = 56.25
\]
- **Upper Bound**:
\[
Q3 + 1.5 \cdot \text{IQR} = 72.5 + 1.5 \cdot 6.5 = 72.5 + 9.75 = 82.25
\]

#### Identify Outliers in Male Height:
- **Lower Bound:** 56.25
- **Upper Bound:** 82.25

Checking the male height data:
- The minimum value is 60.1
- The maximum value is 80.0

Since there are no values below 56.25 or above 82.25, there are **no outliers** in the Male Height dataset.

---

### Female Height Outlier Calculation

#### From previous data:
- **Q1 (Female Height)** = 60.4
- **Q3 (Female Height)** = 63.9

#### Calculate IQR:
\[
\text{IQR} = Q3 - Q1 = 63.9 - 60.4 = 3.5
\]

#### Determine lower and upper bounds:
- **Lower Bound**:
\[
Q1 - 1.5 \cdot \text{IQR} = 60.4 - 1.5 \cdot 3.5 = 60.4 - 5.25 = 55.15
\]
- **Upper Bound**:
\[
Q3 + 1.5 \cdot \text{IQR} = 63.9 + 1.5 \cdot 3.5 = 63.9 + 5.25 = 69.15
\]

#### Identify Outliers in Female Height:
- **Lower Bound:** 55.15
- **Upper Bound:** 69.15

Checking the female height data:
- The minimum value is 54.2
- The maximum value is 75.6

- Values below 55.15:
- 54.2 (outlier)

- Values above 69.15:
- 70.6 (outlier)
- 74.9 (outlier)
- 73.8 (outlier)

#### Outliers in the Female Height dataset:
- **Outliers**: 54.2, 70.6, 74.9, 73.8

### Summary of Outliers
- **Male Height**: No outliers
- **Female Height**: **54.2**, **70.6**, **74.9**, **73.8**

Answer 6

To construct a comparative box plot for the Male and Female Height datasets, we will follow these steps:

1. **Calculate quartiles** and the necessary values for both datasets.
2. **Draw the box plot** for both data sets using the computed values.

### Data Summary

#### Male Height:
- **Minimum:** 60.1
- **Q1:** 66.0
- **Median (Q2):** 69.4
- **Q3:** 72.5
- **Maximum:** 80.0

#### Female Height:
- **Minimum:** 54.2
- **Q1:** 60.4
- **Median (Q2):** 61.65
- **Q3:** 63.9
- **Maximum:** 75.6

### Box Plot Construction

To draw the box plots visually, the following details are to be included:

- **Boxes** represent the interquartile range (IQR), which shows the range between Q1 and Q3.
- **Lines (whiskers)** extend from the box to the smallest and largest values within 1.5 * IQR.
- **A line inside the box** represents the median.

### Box Plot Sketch

While I cannot generate a visual box plot directly as text, I will describe how you would draw one and interpret its shape.

1. **Draw a number line** that covers the range of heights. Make sure it includes all the minimum and maximum values of both datasets.
2. **Draw the box for Male Heights**:
- From Q1 (66.0) to Q3 (72.5)
- Draw a line at the median (69.4).
- Extend the whiskers to the minimum (60.1) and maximum (80.0).

3. **Draw the box for Female Heights**:
- From Q1 (60.4) to Q3 (63.9)
- Draw a line at the median (61.65).
- Extend the whiskers to the minimum (54.2) and maximum (75.6).

### Interpretation of the Box Plot Shapes

1. **Male Height Box Plot**:
- **Shape**: The box will span from 66.0 to 72.5, with the median line at 69.4.
- **Whiskers**: Extend down to 60.1 and up to 80.0 (not a typical distribution).
- **Description**: The male height box plot will likely be skewed to the right (positively skewed) because of the relatively high maximum outlier (80.0), which pulls the whisker out to the right. This suggests a wider range of heights, with a tendency for some exceptionally tall individuals.

2. **Female Height Box Plot**:
- **Shape**: The box will span from 60.4 to 63.9, with the median line at approximately 61.65.
- **Whiskers**: Extend down to 54.2 and up to 75.6, with several possible outliers.
- **Description**: The female height box plot may show a slight positive skew as well because of the few outliers (75.6 and potentially more). However, the distribution is more tightly clustered than the males, indicating less variability in the heights of females.

### Comparative Overview
- The male box plot is wider and exhibits a larger range due to the presence of taller individuals.
- The female box plot is more compact compared to the male plot and shows a smaller IQR indicating less variation in heights among females.

To visualize this, you would typically utilize software or a graphing tool (like Python's Matplotlib or software like Excel/R) to create the box plots accurately. However, the above descriptions showcase how the box plots would appear and be interpreted based on the provided data.

Answer 7

The Empirical Rule, also known as the 68-95-99.7 rule, states that for a normal distribution:

- Approximately **68%** of the data falls within **1 standard deviation (σ)** of the mean (μ).
- Approximately **95%** falls within **2 standard deviations (2σ)** of the mean.
- Approximately **99.7%** falls within **3 standard deviations (3σ)** of the mean.

For the Empirical Rule to be applicable, the data should meet certain conditions:

1. **Normal Distribution**: The data should ideally follow a bell-shaped curve, where most values cluster around the mean and the probabilities for values further away from the mean taper off symmetrically.

2. **No Extreme Outliers**: The presence of outliers can skew the dataset, making it deviate from normality.

3. **Symmetry**: The data should be reasonably symmetric, as significant skewness can indicate a non-normal distribution.

### Analysis of Male and Female Height Datasets

#### Male Height Dataset:
- **Outliers**: The male height dataset did not have any identified outliers, based on the interquartile range method. This suggests that extreme values in this dataset do not drastically deviate from the general trends in the data.
- **Skewness**: The male height box plot suggests a right skew, indicated by a longer upper whisker towards the maximum value (80.0). While there are no extreme outliers, the skewness can still affect the mean, making it higher than the median.
- **Conclusion**: Given the right skew and potential non-normality (despite no extreme outliers), the Empirical Rule may not accurately describe the male height distribution.

#### Female Height Dataset:
- **Outliers**: The female height dataset identified several outliers (54.2, 70.6, 74.9, 73.8), which suggests significant deviations from the central tendency.
- **Skewness**: The female height box plot also suggests slight positive skewness because of the outlier values. The presence of outliers and the compact nature of the bulk of data points (the IQR) indicate that the distribution may not be normal.
- **Conclusion**: The presence of outliers and any potential skewness further precludes the application of the Empirical Rule to the female height dataset.

### Overall Conclusion
In summary, the Empirical Rule can **not** be reliably applied to either the Male or Female Height datasets for the following reasons:

1. **Presence of Outliers**: Although the male dataset had no outliers, the female dataset did. Outliers can distort both the mean and standard deviation, which are core components of the Empirical Rule.

2. **Skewness**: The right skewness of the male height data and the potential skewness in the female height data suggest that the distributions do not closely follow the normal distribution.

3. **Normality**: Neither dataset appears to be perfectly normal based on the skewness and potential outliers, making the assumptions of the Empirical Rule questionable.

Therefore, it is essential to perform additional statistical analysis and/or consider other distribution assessments (e.g., using the Shapiro-Wilk test or visual methods like Q-Q plots) to determine whether these datasets can be adequately described using normal distribution characteristics.