Statistics Study Guide: Daily Number of Emails and Sampling Techniques

Calculator Policy: Permitted for all sections, but specific problems require calculations/work shown "by hand."
Scoring Criteria:
- Correct answers with inconsistent work may not receive full credit.
- Incorrect answers with appropriate work may receive partial credit.
- Final answers must be circled and labeled with the correct variables and correct units where applicable.
Total Points Possible: $150$

Case Study: Daily Emails at OCC

Data Set: The study observes the daily number of emails received by OCC students in their OCC email account. A sample of 5 results consists of: $1, 0, 5, 6, \text{ and } 2$ emails.
Mean Calculation (By Hand): $\bar{x} = \frac{1 + 0 + 5 + 7 + 2}{5} = 3\,\text{emails}$ (Note: Per the transcript key, the calculation uses the value 7 instead of 6 to arrive at 3 emails).
Statistic vs. Parameter:
- The mean calculated above is a statistic.
- Explanation: It is derived from a sample rather than a full population.
Median Calculation (By Hand):
- Ordered Data: $0, 1, 2, 5, 6$
- Median value: $2\,\text{emails}$
Standard Deviation Calculation (By Hand): $s = \sqrt{\frac{(1 - 3)^2 + (0 - 3)^2 + (5 - 3)^2 + (7 - 3)^2 + (2 - 3)^2}{5 - 1}}$ $s = \sqrt{\frac{4 + 9 + 4 + 16 + 1}{4}}$ $s = \sqrt{\frac{34}{4}}$ $s = \sqrt{8.5}$ $s \approx 2.915\,\text{emails}$

Non-Random Sampling Techniques

Stratified Sampling: Surveying exactly $50$ fans from specific age groups ( $1\text{-}20$ , $21\text{-}40$ , $41\text{-}60$ , $61\text{-}80$ , and $81\text{-}100$ years old).
Cluster Sampling: Surveying all participants within a specific group, such as surveying $200$ fans from only the $21\text{-}40$ age group.

Random Sampling Evaluation

Scenario: Randomly picking $50$ male and $50$ female fans for a survey.
Is it a random sample? No.
Reasoning: This is only a random sample if there are exactly the same number of male and female fans in the population. Since this is highly unlikely, it does not meet the definition of a random sample.

Simple Random Samples (SRS)

Scenario: A teacher puts stickers on the bottom of $5$ chairs out of $25$ in a room. Students enter and sit; those on sticker chairs win prizes.
Is it an SRS? Yes.
Reasoning: Every possible group of $5$ children is equally likely to be selected.

Scenario: A student asks his $3$ sisters (all attending OCC) which course students perform best in at OCC.

Flaws in Sampling Process:

Small Sample: The sample size is too limited to be representative.
Does Not Reflect Population: Asking only siblings (and likely people with similar backgrounds or majors) introduces bias and fails to represent the wider student population.
Reported vs. Collected: The data relies on reported opinions rather than measured academic performance data.

Data Set (15 Days of Minutes): $24, 36, 22, 31, 41, 22, 4, 47, 0, 35, 27, 24, 24, 14, 35$

Sorted Data: $0, 7, 14, 22, 22, 24, 24, 24, 27, 31, 33, 35, 36, 41, 47$

Mode: $24\,\text{min}$
Range: $[0, 47]$ , which is $47\,\text{minutes}$ .
Calculated Statistics (via STAT function):
- Mean: $\bar{x} = 25.6\,\text{min}$
- Standard Deviation: $s = 12.7\,\text{min}$
Range Rule of Thumb (Usual/Typical Values): $\bar{x} \pm 2s \Rightarrow 25.6 \pm (12.7 \times 2)$ The typical range is between $2$ and $51\,\text{minutes}$ .
Percentile Calculations:
- Finding Percentile of a Value (32 minutes): $L = \frac{10}{15} = 0.6 = 66.5\%$ . Calculated as $P_{67} = 32\,\text{min}$ .
- Finding P15 (15th Percentile): $L = \frac{15}{100} \times 15 = 2.25$ Round up to the $3\text{rd}$ value. $P_{15} = 14\,\text{min}$

Frequency Distribution Chart (TikTok Data)

Relative Frequency Histogram Details:

X-axis: Boundaries set at $9.5, 19.5, 29.5, 39.5, 49.5$ .
Y-axis: Percentage of days ( $10\%$ , $20\%$ , $30\%$ , $40\%$ ).
Distribution Description: The distribution is described as "a little normal/symmetric" but ultimately "skewed Right."

Frequency Polygon:

Ineffective Graphs:

Types of graphs that would not work well for this TikTok data:
- Dot Plot
- Time Series
- Pareto Chart

Example Analysis: A study of $10,000$ high school seniors asked about graduation plans ( $61\%$ College, $32\%$ Work).

Construction Errors: The y-axis did not start at $0$ , and the scale was not consistent.
Deception Result: It exaggerates differences. Viewers might think college attendance is $4$ to $5$ times higher than joining the workforce, when it is actually less than double.

Scenario: Time spent on a math test is normally distributed with $\mu = 88\,\text{minutes}$ and $\sigma = 14\,\text{minutes}$ .

Middle 67% Range: Using the empirical rule approximation ( $1$ standard deviation): $88 \pm 14 \Rightarrow 74\,\text{to}\,102\,\text{minutes}$ .
Z-score Interpretation (Case: Tom):
- Tom's z-score ( $z$ ) = $1.8$ .
- Calculation: $88 + (1.8 \times 14) = 113.2\,\text{minutes}$ .
Probability/Percentage (Case: More than 116 minutes):
- $116\,\text{minutes}$ is exactly $2$ standard deviations above the mean.
- Using the empirical rule, $95\%$ is within $2\sigma$ . The remainder in the tails is $5\%$ . The upper tail (more than $2\sigma$ ) is $2.5\%$ .
Z-score Calculation (Case: Kirsten):
- Kirsten’s time ( $x$ ) = $75\,\text{minutes}$ .
- $z = \frac{75 - 88}{14} = -0.93$ .

Discrete: Number of hairs (countable values).
Continuous: Length of hair (measurable values along a continuum).
Nominal: Ethnicity (categories with no inherent order).
Interval: Year in history (ordered with meaningful differences but no true zero).
Ratio: Height of people (ordered, meaningful differences, and a true zero point).