NCEA Level 1 Mathematics and Statistics: Interpreting and Applying Information (AS 91946)

Fundamental Requirements of Standard AS 91946

• The AS 91946 1.3 Standard involves interpreting and applying mathematical and statistical information within specific contexts.

• Interpret information: This requires articulating the meaning of numbers, graphs, tables, or displays as they relate to the given context.

• Apply information: This involves using mathematical calculations, identifying specific graph features, and leveraging context to answer questions or formulate specific recommendations.

• Use evidence: High-quality responses must include numerical evidence, including but not limited to:

  • Percentages
  • Medians
  • Quartiles
  • Ranges
  • Trend values
  • Calculated costs

• Discuss confidence: Learners must explain the reliability of their answers. Factors to consider include:

  • Sample size
  • Sampling method used
  • Variability within the data
  • Underlying assumptions made during analysis

• Think beyond one number: Excellence-level answers connect various pieces of evidence, consider the context and its limitations, and apply real-world judgment rather than relying on a single data point.

Analysis and Interpretation of Statistical Displays

• Dot plot with box plot:

  • Key features to discuss: Centre, shift, shape, spread, outliers, and overlap.
  • Useful sentence starter: "The median for [Variable A] is [Value], compared with [Variable B]. This suggests…"

• Time series graph:

  • Key features to discuss: Overall trend, repeating patterns, seasonal patterns, unusual features, and predictions.
  • Useful sentence starter: "Overall, the data shows… A repeating pattern is…"

• Scatter plot:

  • Key features to discuss: Direction, form, strength, variation, outliers, and confidence in predictions.
  • Useful sentence starter: "There is a [Strength/Direction] relationship between [Variable X] and [Variable Y]."

• Cost table:

  • Key features to discuss: Goods and Services Tax (GST), discounts, shipping costs, specific conditions, and comparing costs such as one item versus two items.
  • Useful sentence starter: "The total cost is [Value] because…"

• Survey infographic:

  • Key features to discuss: Percentages, sample size, subgroup patterns, potential bias, and generalisability to the wider population.
  • Useful sentence starter: "I am [Level of Confidence] because the survey…"

Comparing Data Distributions

• Centre: Compare the medians or means of the datasets and explain what these differences signify in the real-world context.

• Shift: Determine which distribution is generally positioned higher or lower on the scale than the other.

• Shape: Describe specific visual characteristics such as clustering (where data points group together), skew (the direction of the tail), gaps in the data, and any unusually high or low values (potential outliers).

• Spread: Compare the range or the Interquartile Range (IQR). A larger spread indicates higher variability within the dataset.

• Conclusion: Formulate an answer to the specific question based on the context, rather than simply listing the visual features of the graph.

Time Series Analysis and Predictions

• Trend: Refers to the long-term direction in which the data is moving (upward, downward, or stationary).

• Repeating pattern: Refers to regular rises and falls in the data that are often linked to specific cycles such as seasons or financial quarters.

• Unusual feature: Refers to a specific data point or a period of time where the data deviates significantly from the established trend or pattern.

• Prediction: Predictions should be made by utilizing the overall trend and the most recent patterns.

• Confidence in Predictions: Confidence is generally lower if the data demonstrates:

  • High variability
  • Unusual events
  • A noticeable change in the established trend

Bivariate Relationships and Predictions

• Direction:

  • Positive: As one variable increases, the other variable also tends to increase.
  • Negative: As one variable increases, the other variable tends to decrease.

• Strength: A "strong" relationship exists when data points are positioned very close to a clear pattern. A "weak" relationship exists when data points are scattered widely.

• Form:

  • Linear: The pattern of data points roughly follows a straight line.
  • Nonlinear: The pattern of data points is curved.

• Prediction: Predictions are made by reading from the trend line. Uncertainty must be discussed by evaluating the scatter of points, presence of outliers, and whether the input value falls within the existing data range (interpolation vs. extrapolation).

Evaluating Reliability and Limitations

• Reliable evidence: Characterized by an appropriate sample size, a clearly defined data source, and a fair/randomized sampling method.

• Reduced reliability: A result may be less reliable if the sample data is sourced from:

  • Only one website
  • Use of a single specific group
  • Voluntarily provided data (vulnerable to self-selection bias)
  • An unknown or undefined population

• Recommendations: When making recommendations, always state assumptions and limitations. These may include:

  • Future changes in costs
  • Individual personal preferences
  • Changes in future needs
  • Missing or incomplete information

• Excellence-level performance: Usually requires connecting several disparate pieces of evidence and providing a clear explanation of uncertainty.

Writing Frames for Statistical Communication

• To Describe: "The graph shows [Feature] in the context of [Context]."

• To Compare: "[Item A] is higher than [Item B] because [Numerical Evidence]. This means [Contextual Impact]."

• To Calculate: "I used [Specific Method/Value] because [Reason]. The value is [Result], so [Implication]."

• To Predict: "I predict [Outcome] because [Evidence from Trend/Pattern]. I am [Level of Confidence] because [Reason]."

• To Recommend: "I recommend [Choice]. The evidence is [Data Point]. However, [Limitation or Assumption]."

Practice Resource 1: Adidas and Nike Shoe Prices

• Resource 1A Data: In a sample of $126$ Nike shoes, $6$ of them cost more than $500\,\text{dollars}$.

• Percentage Calculation:

  • Formula: $\frac{6}{126} \times 100$
  • Interpretation: This represents the proportion of high-end/luxury pricing within the Nike sports shoe distribution.

• Comparing Distributions: Analysis must include two features (Centre, Shift, Spread, or Shape) and use specific numerical evidence from Resource 1A.

• Reliability Concerns: Resource 1A may not be fully reliable for general sports shoe comparisons because it might be limited to specific retailers, might not account for all styles, or the sample size of $126$ may not represent the entire global inventory of both brands.

Practice Resource 1B: Spending on Clothing and Footwear

• Trend Analysis: Identify one long-term direction in spending and support it with data.

• Pattern Analysis: Identify one repeating seasonal or quarterly cycle in spending with supporting evidence.

• Unusual Feature Analysis: Identify a specific point of deviation in the spending data.

• Prediction and Confidence: Make a future expenditure prediction based on current trends and evaluate confidence based on data variability.

Practice Resource 2: Shoe Size, Price, and Purchasing

• Resource 2A Characteristics (Size and Price):

  • Describe two features of the relationship (e.g., strength of correlation between size and cost).
  • Use the trend line to estimate the cost for a size $6$ shoe.
  • Discuss confidence based on the scatter around the size $6$ mark.

• Resource 2B (Cost Comparison):

  • Recommendations must factor in: Base prices, GST, discounts, and shipping conditions.
  • Scenario: Choosing the best option for one pair of size $7$ football shoes.
  • Future considerations: If a person (e.g., Ricky) expects their shoe size to increase by next season, the recommendation must be re-evaluated using data from Resource 2A (price trends for larger sizes) and Resource 2B (current availability/costs).

Practice Resource 3: Running Shoe Survey

• Survey Parameters:

  • Sample Size: $2571$ people.
  • Source: Visitors of the "Running Shoes Guru" website.

• General Buying Habits (Percentage of $2571$ people):

  • Run in the morning: $53\,\%$
  • Spend $80\,\text{dollars}$ to $120\,\text{dollars}$: $47\,\%$
  • Consider comfort: $38\,\%$
  • Consider personal experience: $26\,\%$
  • Consider brand: $7\,\%$

• Overall Brand Preferences:

  • Asics: $22\,\%$
  • Nike: $20\,\%$
  • Adidas: $19\,\%$
  • New Balance: $17\,\%$
  • No favourite: $14\,\%$
  • Other makes: $8\,\%$

• Brand Preference by Age (Excluding "Other" brands):

  • Under 20: Nike leads significantly at $46\,\%$.
  • 21 to 30: Nike at $33\,\%$, Asics at $22\,\%$.
  • 31 to 40: Asics and Nike tied at $24\,\%$.
  • 41 to 50: Nike at $25\,\%$, Asics at $21\,\%$, Adidas and New Balance follow.
  • 51 to 60: Nike at $27\,\%$, Asics at $21\,\%$.
  • Over 60: Nike at $26\,\%$, Adidas at $22\,\%$, Asics at $21\,\%$.

• Statistical Estimates:

  • To find the approximate number of morning runners: $0.53 \times 2571 \approx 1363\, \text{people}$.

• Confidence in Survey Brand Popularity:

  • While Asics is the top brand in the overall survey at $22\,\%$, confidence in calling it the "most popular" generally must be tempered by the fact that the sample was taken from a specific niche website (Running Shoes Guru), which may not represent all runners.

• Applied Recommendation Example (Axel):

  • Profile: $45$ years old, runs $50\,\text{km}$ per week.
  • Recommendation: Based on age group data ($41$ to $50$), Nike ($25\,\%$) or Asics ($21\,\%$) are statistically preferred. Estimated cost for a dedicated runner often falls in the $80\,\text{dollars}$ to $120\,\text{dollars}$ range ($47\,\%$) based on the spending habit statistics.

Final Response Checklist

• Ensure the question is addressed directly.
• Incorporate numerical evidence from provided resources.
• Interpret the statistics within the specific context of the problem.
• Explicitly state any relevant limitations, assumptions, or levels of confidence.
• Ensure conclusions and recommendations are logically derived from the presented evidence.