Statistics, Probability & Power BI - Evaluation (Actual Paper, Set 2)

Conditional probability restricts attention to the baskets where the condition holds: P(tea | milk) = P(tea and milk) / P(milk) = (100/400) / (160/400) = 100/160 = 0.625. Dividing 100 by all 400 baskets gives the joint probability, 160/400 is just the probability of milk, and 60/160 = 0.375 is the share of milk baskets that have no tea.

Bayes' theorem gives the posterior P(default | flagged) = P(flagged | default) x P(default) / P(flagged) = 0.75 x 0.04 / 0.12 = 0.25. The 75% figure is the likelihood, which answers the reverse question; 4% is the prior before any evidence is seen; and 9% results from multiplying the likelihood by the evidence term instead of dividing.

When hypotheses are only being compared, the posterior is proportional to likelihood x prior because the denominator P(evidence) is constant across the comparison. The products are 0.50 x 0.02 = 0.010 for Promotional, 0.30 x 0.05 = 0.015 for Transactional and 0.20 x 0.06 = 0.012 for Phishing, so Transactional wins. The highest likelihood (Phishing) or the highest prior (Promotional) alone does not decide the result, and the evidence term does not differ between categories.

Five values with a mean of 40 must total 200, and the known four total 160, so the fifth value is forced to be 40 (the deviations from the mean always sum to zero). Once the sample mean is computed from the data, only n - 1 = 4 observations can vary independently while the last is constrained, which is why the sample variance divides by the degrees of freedom n - 1 instead of n. Treating the fifth value as free contradicts the fixed mean, and keeping five free values would not justify an n - 1 divisor.

The z-score z = (x - μ) / σ expresses each value in units of its own standard deviation: age gives (45 - 35) / 5 = 2 and income gives (100,000 - 80,000) / 20,000 = 1. So the applicant is more extreme on age even though the raw income gap looks far larger, which is exactly why standardising puts features with different scales on a common scale. Standardising does not force every value to the same z, and the common scale removes the unit problem.

When the mean is larger than the median, a handful of very large values are dragging the average to the right of the typical value, which is the pattern of right-skewed data with a long tail towards the right. A left-skewed set would show the mean below the median, and a symmetric set would show the two close together. Salary is numeric, so the mean and median are both valid here and the mode is not the only usable measure.

A continuous variable has infinitely many possible values, so the probability of one exact value is zero and probability is found as the area under the curve between two points. Total probability is 1 in both cases: a PMF sums to 1 over discrete outcomes and a PDF integrates to 1 over its range. A PDF value is a density, not a probability, and the peak of the normal curve at the mean is the highest density, not a non-zero single-value probability.

26 minutes is one standard deviation below the mean and 38 minutes is two standard deviations above it. Because the distribution is symmetric, half of the 68% band (34%) lies between 26 and 30 and half of the 95% band (47.5%) lies between 30 and 38, giving 81.5%, which is about 82%. 68% would cover only 26 to 34 minutes, 95% would cover 22 to 38 minutes, and 84% is the share of deliveries above 26 minutes without any upper cut-off at 38.

The standard error is σ/√n, so it shrinks with the square root of the sample size. Halving it from 2.0 to 1.0 requires √n to double, which means n must grow by a factor of four, from 400 to 1,600. Doubling to 800 would only reduce the standard error by a factor of about 1.41, and the other two options rely on relationships that the formula does not contain.

Picking whoever is easiest to reach is convenience sampling, and it is highly biased when the goal is to say something about students across the whole country because one location and one day cannot represent them. Stratified sampling needs groups that already exist in the population and a proportional draw from each. Systematic sampling needs a fixed rule such as every kth person. Simple random sampling needs every student in the population to have an equal chance of selection, which students in other cities do not.

By the central limit theorem the distribution of sample means approaches a normal distribution regardless of the shape of the original population, so the histogram of means is roughly bell-shaped and centred near the population mean. Its spread is the standard error σ/√n = 40/√64 = 5, far narrower than the population's 40. The other options either keep the population's skew or keep its wider spread, and averaging changes both.

The binomial distribution needs binary outcomes, a fixed number of trials, a constant success probability and independent trials; 15 loans with the same 4% default probability and independent behaviour meet all four. A count of calls known only through an average is the situation for the Poisson distribution, a delivery time is a continuous variable that needs a density over ranges, and visitors whose chances change after each purchase break the independence and constant-probability requirements.

The decision rule rejects H₀ when the p-value is smaller than the level of significance. Since 0.03 is below both 0.05 and 0.10, H₀ is rejected at those levels, but 0.03 is not below 0.01, so H₀ is not rejected at that stricter level. A low p-value is strong evidence against H₀; it does not prove that H₁ is true.

A paired test applies when the same group is measured twice or when matched items are compared, which covers the same patients before and after medication and the same books priced offline and online. Two different sets of phones are independent samples and need a two-sample independent t-test, and three plants involve more than two groups, which calls for ANOVA.

Rejecting a null hypothesis that is actually true is a Type I error, the false-positive case of declaring an innocent person guilty, and its probability is the level of significance α. A Type II error is the opposite mistake of failing to reject a false null hypothesis, and its probability is β, where power is 1 - β.

A chi-square test checks for an association between two categorical variables, which is exactly the question of whether return status is independent of city. One-way ANOVA compares the means of a numeric variable across groups, a paired t-test needs the same subjects measured under two conditions, and a one-sample t-test compares one sample mean with a benchmark value; none of these tests whether two categorical variables are related.

Five cities form 5 x 4 / 2 = 10 distinct pairs, so pairwise testing would need ten t-tests, and running many tests inflates the chance of a false positive. A single one-way ANOVA compares the means of more than two groups at once and avoids this multiple comparison problem. Counting 5 or 20 tests miscounts the pairs, and chi-square is a test for two categorical variables, not for comparing mean delivery times.

The confusion matrix is TP = 16, FP = 24, FN = 4 and TN = 956, so accuracy is 972/1000 = 97.2%, precision is 16/40 = 40% and recall is 16/20 = 80%. With only 2% fraud, a model that never flags anything would already reach 98% accuracy, so accuracy is misleading on imbalanced data and precision and recall tell the real story: the model catches most fraud but most of its flags are false alarms. Reading recall as 40% and precision as 80% swaps the two metrics, trusting accuracy because the dataset is large ignores the imbalance, and calling the classes balanced is wrong since only 2% of transactions are fraudulent.

Fields that look numeric, such as postal codes and order IDs, are dimensions when they are used to slice or group the data, and the semantic model applies cosmetic transformations such as renaming fields once so that they flow to all downstream visualizations. A worksheet is a single chart and a dashboard combines several worksheets, so calling a worksheet a collection of dashboards reverses the relationship. A fresh query on every view describes a live connection, whereas an extract serves stored snapshots refreshed on a schedule.

Power Query cleans, transforms and structures data before the dashboard is created, and promoting the first row to headers and find-and-replace on values are both of that kind. Summing a column into a measure and counting distinct customers are calculations done with DAX after the data is loaded.

Desktop-authored BI tools such as Tableau and Power BI have a desktop authoring app for building charts and dashboards, server or cloud infrastructure that manages data connections and automatic updates in the background, and a public viewer through which stakeholders explore dashboards without technical knowledge. Re-running a notebook for every viewer is the manual Python approach that BI tools replace, not one of their components.

A calculated column is computed per row: O1 gives 4 x 500 - 400 - 100 = 1,500 and O2 gives 2 x 300 - 150 - 60 = 390. Total Profit sums them to 1,890, and the margin measure divides total profit by total sales: 1,890 / 2,600 is about 72.7%. Averaging the two order margins (75% and 65%) to get 70% is a different calculation, because the measure aggregates profit and sales first and divides afterwards.

Measures are the numeric quantities that get aggregated with operations such as sum or average and appear as the values on an axis, and profit is exactly that. Dimensions slice and organise the data into chart headers, axis labels and groups; region, postal code and order ID all play that role, even though postal code and order ID look numeric.

SUMX works row by row and then aggregates: Choco gives 100 x 8 = 800, Oatmeal gives 50 x 6 = 300 and Almond gives 40 x 15 = 600, so the total is INR 1,700. INR 1,837 comes from multiplying total units by an averaged margin, INR 2,050 is only the total cost, and INR 3,750 is only the total revenue.

Extract is the sensible default: a once-a-day revenue view tolerates a scheduled lag, renders fast from stored snapshots and spares the shared warehouse from a query on every view. A live connection is justified only where real-time data is mandatory, which is the failure wall. Making both live hammers the warehouse and raises cost, making both extracts leaves the wall stale, and reversing the roles puts the real-time need on the lagging extract.

Direct labelling removes the legend, which forces readers to juggle between legend and graph, and light gray gridlines used sparingly keep only the ink that helps read specific values. 3D effects are hard to interpret on a 2D surface and a heavy border is unnecessary framing, so both add decoration instead of data.

Meaning should never be carried by colour alone, so adding symbols such as plus and minus signs (or patterns and different line styles) lets every viewer read the gain or loss. A rainbow palette lacks contrast and hierarchy, a stronger red-green pair is still a red-green pair, and relocating the legend does nothing for the colour-only encoding.

A pair plot plots every numerical variable against every other one without the user choosing, which makes it the quick tool for exploratory analysis, whereas a facet grid needs the analyst to choose which variable to split on and so serves a specific, predetermined comparison such as one relationship across four regions. A pair plot of five numeric columns is a 5 x 5 grid of 25 panels (the diagonal shows distributions and the mirror panels repeat each other), not ten, and the automatic every-pair behaviour belongs to the pair plot, not the facet grid.

In Plotly Express, color colour-codes points by a categorical column, size sets the marker size from a numerical variable (here usage_c), and animation_group keeps each object's identity across animation frames while animation_frame animates the figure over the values of month. hover_name only controls the name displayed when a user hovers over a point; it does not colour anything.

A pair plot plots every numerical variable against every other numerical variable, so the four numeric columns give a 4 x 4 grid of 16 panels and the categorical columns only supply the colour grouping through hue. The diagonal shows each variable's distribution rather than a scatter plot, and corner=True removes the redundant mirror panels above it. That leaves the 4 diagonal panels plus 6 unique lower-triangle scatters, which is 10 panels.