VCAA General Mathematics Data analysis, probability and statistics

This option incorrectly uses the difference between the two given values ($3\sigma = 21.3$) as the standard deviation instead of dividing by 3 to find $\sigma$.

While the standard deviation of 7.1 is correct, the mean is miscalculated. The mean should be found by adding the standard deviation to 157.6.

This option incorrectly calculates the standard deviation as $2\sigma = 14.2$ and subsequently miscalculates the mean.

This incorrect value likely results from an arithmetic error when calculating the y-intercept or applying the residual formula $e = y - \hat{y}$.

This answer is incorrect and may stem from missing the negative sign on the correlation coefficient, which would incorrectly change the slope and intercept of the least squares line.

This incorrect residual could result from reversing the ratio of standard deviations when calculating the slope ($s_x/s_y$ instead of $s_y/s_x$), leading to an incorrect line equation.

This is the raw data value for Day 4, not the calculated moving average.

This value is incorrect; the average of the second window of data points (10, 18, and 32) is 20, not 25.

This is the raw data value for Day 3, not the calculated moving average.

Neither variable asks for a specific numerical measurement, such as an exact dollar amount. Since the responses are groups or labels, they are not numerical variables.

While payment preference is categorical, budget preference is also categorical because it groups responses into ranges ('less than ＄50' or 'more than ＄50') rather than asking for an exact numerical value.

Payment preference ('cash' or 'electronic') is clearly a categorical variable, not numerical. Budget preference is also categorical, making this option entirely incorrect.

A negative association requires a correlation coefficient less than zero ($r < 0$), but the given value is $0.23$.

A strong positive association typically corresponds to an $r$ value closer to $1$ (e.g., $r > 0.7$), whereas $0.23$ represents a weak relationship.

This option describes a correlation coefficient close to $-1$, but the given value is positive and indicates a weak relationship.

This reverses the variables; $R^2$ measures the proportion of variation in the response variable ($y$) explained by the explanatory variable ($x$), not the variation in $x$ explained by $y$.

$R^2$ measures the proportion of the variation in the observed response values ($y$), not the predicted outcomes, that is explained by the model.

This refers to the variation in the explanatory variable ($x$), whereas $R^2$ specifically measures the explained variation in the response variable ($y$).

This would correspond to a seasonal index of 0.25 (calculated as $1.0 - 0.75$), not 0.75.

This would correspond to a seasonal index of 1.25 (calculated as $1.0 + 0.25$), meaning sales are above average.

This would correspond to a seasonal index of 1.75 (calculated as $1.0 + 0.75$), meaning sales are significantly above average.

Incorrect. If $\mu = 45$ and $\sigma = 6$, then $\mu + \sigma = 51$, which would mean only 16% of eggs weigh more than 51 grams, contradicting the given 97.5%.

Incorrect. These values do not satisfy the system of equations derived from the 68-95-99.7% rule: $\mu + \sigma = 96$ and $\mu - 2\sigma = 51$.

Incorrect. If $\mu = 73.5$ and $\sigma = 11.25$, then $\mu + 2\sigma = 96$, which would mean only 2.5% of eggs weigh more than 96 grams, not the stated 16%.

This is an example of univariate data because it involves only one variable (the rating) for each observation.

This is univariate data because it records only a single variable (the count of cars) at a specific location.

This is univariate data because it measures only one variable (time spent) for each person observed.

Incorrect. A 25% reduction would mean multiplying the actual sales by 0.75, which does not match the required deseasonalizing factor of $1/1.75 \approx 0.57$.

Incorrect. This is the percentage that the deseasonalized sales represent of the actual sales ($1/1.75 \approx 57\%$), rather than the percentage by which the actual sales must be reduced.

Incorrect. While a seasonal index of 1.75 means sales are 75% above the seasonal average, reversing this increase requires dividing by 1.75, not subtracting 75%.

This describes a discrete variable, which is defined by having a countable number of possible values, rather than a confounding variable.

A value that remains constant is simply a constant or a controlled variable, whereas a confounding variable varies and affects the outcome.

This describes the explanatory (or independent) variable, which is the specific factor the researcher is studying to see if it causes a change.