Covers Lessons 6–10: measures of centre and spread, representing data, comparing data sets, bivariate data analysis, and regression analysis.
This assessment follows the former Statistical Analysis sequence, including earlier probability and data topics. Use it for 2017-syllabus / 2026 HSC revision or optional extension. For the 2024 syllabus / first HSC 2027, use the ten current focus-area lessons and their core practice.
Assessment
Select the best answer for each question.
A data set has mean 15 and standard deviation 4. If 3 is added to every value, what is the new standard deviation?
In a histogram with unequal class widths, what does the area of each bar represent?
A student scores 78 in a test with mean 70 and SD 8. What is their z-score?
Pearson's correlation coefficient $r = −0.85$ indicates:
Which of the following is a valid conclusion from a strong correlation between two variables?
The regression line $\hat{y} = 20 + 5x$ has slope $b = 5$. This means:
A residual plot shows a clear curved pattern. What should you conclude?
Predicting $y$ for an $x$ value well outside the original data range is called:
Short Answer
The test scores of 10 students are: 52, 58, 62, 65, 68, 70, 72, 75, 80, 88. (a) Find the mean, median, and interquartile range. (b) Identify any outliers using the 1.5 × IQR rule. (c) A new student scores 95. Recalculate the mean and explain why the median might be a better measure of centre for this updated data set.
For a data set relating hours studied ($x$) to exam scores ($y$): $\bar{x} = 6$, $s_x = 2$, $\bar{y} = 72$, $s_y = 12$, and $r = 0.75$. (a) Find the equation of the least-squares regression line. (b) Predict the exam score for a student who studied 8 hours. Is this interpolation or extrapolation? (c) Calculate the residual if a student who studied 8 hours actually scored 88.
A study finds $r = 0.92$ between monthly chocolate consumption per capita and number of Nobel Prize winners per capita across countries. (a) Describe the scatter plot you would expect to see. (b) A newspaper headline claims: "Eating chocolate makes you smarter." Identify and explain three statistical errors in this claim. (c) Propose a more likely explanation for this correlation.
Q1: C, Adding a constant shifts the mean but leaves spread unchanged. SD remains 4.
Q2: B, With unequal widths, bar height = frequency density, and area = frequency.
Q3: C, $z = (78 - 70) / 8 = 1.0$.
Q4: B, $|r| = 0.85$ is strong; the negative sign indicates a negative relationship.
Q5: C, Correlation only shows association. Causation, linearity, and confounding cannot be concluded from $r$ alone.
Q6: B, The slope is the change in $\hat{y}$ per 1-unit increase in $x$.
Q7: B, A curved residual pattern suggests the true relationship is non-linear.
Q8: B, Extrapolation is predicting outside the data range and is unreliable.
Q9 (3 marks): (a) Mean = $\frac{690}{10} = 69$ [0.5]. Median = $\frac{68+70}{2} = 69$ [0.5]. $Q_1 = 62$, $Q_3 = 75$, IQR = 13 [0.5]. (b) Lower fence = $62 - 19.5 = 42.5$; Upper fence = $75 + 19.5 = 94.5$. No outliers in original data [0.5]. (c) New mean = $\frac{785}{11} \approx 71.4$ [0.5]. The median (now 70) is better because the mean is pulled up by the high outlier (95), while the median is robust to extreme values [0.5].
Q10 (3 marks): (a) $b = 0.75 \times (12/2) = 4.5$ [0.5]. $a = 72 - 4.5(6) = 72 - 27 = 45$ [0.5]. $\hat{y} = 45 + 4.5x$ [0.5]. (b) $\hat{y}(8) = 45 + 4.5(8) = 45 + 36 = 81$ [0.5]. This is interpolation (8 is within the data range, assuming $x$ ranges around 6 ± 2 SD) [0.5]. (c) Residual = $88 - 81 = 7$ [0.5].
Q11 (3 marks): (a) Tight cluster of points rising from left to right, strong positive linear trend [0.5]. (b) Three errors: (1) Correlation does not prove causation, no mechanism shown. (2) Observational data, confounding variables (wealth, education, research funding) likely explain both. (3) "Makes you smarter" implies a direct effect, but temporal order and experimental evidence are absent. (4) Ecological fallacy, country-level averages may not apply to individuals [1.5]. (c) Wealthier countries consume more chocolate AND invest more in research, producing more Nobel winners. Chocolate consumption is a marker of wealth, not a cause of intelligence [1].