Statistics for Data Science I · Quiz 1 · Sep 2025

**1. Convert the relative frequency to a count:** There are 10 ratings, so a relative frequency of 0.30 means rating 9 occurs 0.30×10=3 times. **2. Count the known occurrences:** The listed nine known ratings are 7,8,6,9,7,5,8,9,8. Exactly two are 9, so the missing rating must provide the third occurrence. **Answer: B — 9.** A: **Incorrect:** Choosing 7 leaves only two 9s and a relative frequency of 0.20. B: **Correct:** Choosing 9 produces three 9s among ten ratings. C: **Incorrect:** Choosing 8 leaves the relative frequency of 9 at 0.20. D: **Incorrect:** Choosing 6 leaves only the original two 9s.

**1. Compare proportions within each year:** First-year students choose online in 40 of 60 cases, or 66.67%. Second-year students choose online in 20 of 60 cases, or 33.33%. **2. Interpret the difference:** If the variables were independent in this table, the learning-mode proportions would be the same in both year groups. They differ, so mode of learning and year of study are associated in these observed data. **Answer: A — Yes.** **Convention note:** This describes association in the given contingency table, not causation. A formal population-level significance test is a separate inferential question. A: **Correct:** The conditional online-learning proportions differ substantially between the two year groups. B: **Incorrect:** Equal row and column totals do not establish independence; the conditional proportions differ.

**1. Calculate the total sales:** The 30-member mean is Rs.60,000, so \(T=30(60{,}000)=1{,}800{,}000\) rupees. **2. Subtract the known groups:** The top ten contribute \(10(90{,}000)=900{,}000\); the bottom ten contribute \(10(30{,}000)=300{,}000\). Thus the middle ten contribute \(1{,}800{,}000-900{,}000-300{,}000=600{,}000\). **3. Divide by the middle group’s size:** \[\bar x_{middle}=\frac{600{,}000}{10}=60{,}000\text{ rupees}.\] **Answer: D — ₹60,000.** A: **Incorrect:** It would give only Rs.480,000 for the middle group, below the required Rs.600,000. B: **Incorrect:** It would give only Rs.450,000 for the middle group. C: **Incorrect:** It repeats the bottom-group average rather than using the middle-group total. D: **Correct:** Ten members averaging Rs.60,000 provide the required remaining Rs.600,000.

**1. Identify what meaningful differences require:** Interval-scale measurements have equal-sized units, so differences and arithmetic adjustments in those units are meaningful. Ratio scales retain this property and additionally have a meaningful absolute zero. **2. Exclude labels and ranks:** Nominal values identify categories; ordinal values order them without specifying equal gaps. Arithmetic on their numeric codes does not have a general quantitative interpretation. **Answer: A, D — A and D — Ratio and Interval.** **Convention note:** For interval data, meaningful equal-unit arithmetic does not make ratios of absolute readings meaningful. For example, 20°C is not twice as hot as 10°C; a 10-degree difference is meaningful. A: **Correct:** Ratio measurements have equal units and a true zero, permitting meaningful differences and ratios. B: **Incorrect:** Category labels do not provide meaningful numerical additions or differences. C: **Incorrect:** Rank codes do not establish equal intervals between adjacent categories. D: **Correct:** Interval measurements have equal units, so differences are meaningful even though absolute ratios depend on the zero point.

**1. Distinguish the purposes:** Descriptive statistics summarizes the data actually observed. Inferential statistics uses sample evidence, together with a statistical model or sampling assumptions, to reason about a wider population. **2. Identify examples:** Means, medians, standard deviations, tables and charts are descriptive summaries. Estimating population characteristics or making population predictions from a sample is inferential. Either branch can analyze categorical or numerical data. **Answer: B, D — B and D.** A: **Incorrect:** It reverses the usual roles of descriptive summaries and probability-based inference. B: **Correct:** Generalizing from a sample to a population is the purpose of statistical inference. C: **Incorrect:** Categorical frequencies, proportions and associations are also statistical subjects. D: **Correct:** These numerical measures summarize the center and spread of observed data.

**1. Distinguish data collection over units and time:** Cross-sectional data compare different units at one time or reference period. Time-series data follow observations across multiple ordered times. **2. Distinguish measurement scales:** Nominal categories are unordered labels. Ratio measurements have equal units and a true zero, supporting meaningful differences and ratios. **Answer: A, D — A and D.** A: **Correct:** It correctly describes observations across multiple units at one reference time. B: **Incorrect:** Time-series data require observations across times, not a single time point. C: **Incorrect:** Meaningful category ranking is an ordinal property, not a nominal one. D: **Correct:** A true zero and equal units permit both difference and ratio comparisons.

**1. Identify the centered sums:** The data give \(S_{xx}=\sum(X_i-\bar X)^2=100\), \(S_{yy}=400\), and \(S_{xy}=\sum(X_i-\bar X)(Y_i-\bar Y)=160\). **2. Apply Pearson’s formula:** \[r=\frac{S_{xy}}{\sqrt{S_{xx}S_{yy}}}=\frac{160}{\sqrt{100(400)}}=\frac{160}{200}=0.8.\] **3. Interpret and format:** The association is positive and relatively strong. The requested one-decimal answer is 0.8. Because centered sums are already supplied, the means 8 and 60 need not be substituted again. **Answer: 0.8.**

**1. Recover the total sample size:** Action has frequency 36 and relative frequency 0.30, so \(N=36/0.30=120\). **2. Fill the missing categories:** Comedy \(a=0.25(120)=30\), Drama \(b=0.20(120)=24\), and Others \(e=0.10(120)=12\). The remaining Horror proportion is \(1-0.30-0.25-0.20-0.10=0.15\), giving \(c=18\). **3. Add the requested frequencies:** \[a+b+c=30+24+18=72.\] These are counts, rather than the sum of the relative frequencies. **Answer: 72.**

**1. Read the paired observations:** Temperature X is 20,25,30,45 and ice creams sold Y is 30,55,70,85. Their population means are 30 and 60. **2. Multiply paired deviations:** The products are \[(-10)(-30)=300,\quad(-5)(-5)=25,\quad(0)(10)=0,\quad(15)(25)=375.\] Their sum is 700. **3. Use the population denominator:** \[\operatorname{Cov}(X,Y)=\frac{700}{4}=175.\] Dividing by 3 would calculate sample covariance, which is not requested. **Answer: 175.**

**1. Sort the step counts:** In thousands, the ordered data are 8,15,17,19,21,23,25,35. **2. Use medians of the lower and upper halves:** The lower half is 8,15,17,19, giving \(Q_1=(15+17)/2=16\). The upper half is 21,23,25,35, giving \(Q_3=(23+25)/2=24\). Thus \(IQR=24-16=8\). **3. Calculate the upper fence:** \[Q_3+1.5IQR=24+1.5(8)=36.\] No observation exceeds 36 thousand steps; even the maximum, 35, is below it. Hence zero users are marked. **Answer: 0.** **Convention note:** Quartiles are not uniquely defined for finite samples. The stated answer uses the median-of-halves convention, consistent with this key. Inclusive linear interpolation instead gives Q1=16.5, Q3=23.5 and an upper fence of 34, which would mark the value 35 and produce one outlier. The question does not specify which convention to use.

**1. Obtain the missing sum:** Let the unknown observations be a and b. The total is \(4(5)=20\), so \(a+b=20-4-6=10\). **2. Obtain the missing sum of squares:** Population variance satisfies \(\sigma^2=\sum x_i^2/n-\bar x^2\). Therefore \[\sum x_i^2=4(2.5+5^2)=110,\qquad a^2+b^2=110-4^2-6^2=58.\] **3. Extract the product:** \[(a+b)^2=a^2+b^2+2ab\Rightarrow100=58+2ab\Rightarrow ab=21.\] The two observations can indeed be 3 and 7. **Answer: 21.**

**1. Compare the spread rather than just the center:** Both branch means are 36 parcels/day, but Branch A has standard deviation 0 and Branch B has standard deviation 8. **2. Interpret zero standard deviation:** A sum of squared deviations can be zero only when every observation equals the mean. Therefore A delivered 36 on each of the five observed days, whereas B’s observations vary. **3. State the consistency conclusion:** Lower observed variability means A is more consistent over these days, despite equal average output. **Answer: B, C — B and C.** **Convention note:** The claim about A delivering exactly 36 every day refers to the observed five-day dataset; it does not guarantee all future days will be identical. A: **Incorrect:** Equal means describe equal average output, not equal day-to-day variability. B: **Correct:** A has lower standard deviation and therefore less observed variation. C: **Correct:** Zero standard deviation forces every observed A value to equal 36; B’s positive standard deviation indicates variation. D: **Incorrect:** Consistency concerns spread and cannot be determined from the mean alone.

**1. Identify the recording transformation:** The original data are 30,45,25,35,45 and the reported data are 60,90,50,70,90. Every value was multiplied by 2. **2. Apply the variance scaling rule:** The original standard deviation is 8, so its variance is \(8^2=64\). For \(Y=2X\), \[\operatorname{Var}(Y)=2^2\operatorname{Var}(X)=4(64)=256.\] **3. Verify directly as a population variance:** The reported mean is 72. Squared deviations are 144,324,484,4,324, totaling 1280; dividing by 5 gives 256. **Answer: 256.**

**1. Add category sales across the three years:** Action totals 20+30+25=75; Drama totals 40+50+35=125; Comedy totals 20+10+15=45; Sci-fi totals 20+10+25=55. Overall sales total 300. **2. Divide each category total by all sales:** \[\text{Action}=25\%,\quad\text{Drama}=41.67\%,\quad\text{Comedy}=15\%,\quad\text{Sci-fi}=18.33\%.\] **3. Identify the largest share:** Drama has 125 of 300 sales and therefore the highest overall share. Aggregating counts first avoids treating unequally sized years as equally weighted; here every year happens to total 100. **Answer: B — Drama — 41.67%.** A: **Incorrect:** Its 25% share is below Drama’s 41.67%. B: **Correct:** Its 125 of 300 sales give the highest share, approximately 41.67%. C: **Incorrect:** Its 15% share is the smallest among these categories. D: **Incorrect:** Its 18.33% share is below both Action and Drama.

**1. Calculate annual shares:** Each year totals 100 sales. Thus Action/Drama shares are 20%/40% in 2021, 30%/50% in 2022 and 25%/35% in 2023. **2. Calculate absolute share gaps:** \[|20-40|=20,\qquad|30-50|=20,\qquad|25-35|=10\] percentage points. **3. Take the maximum:** The greatest gap is 20 percentage points, attained in both 2021 and 2022. The options display this share difference with a percent sign. **Answer: D — 20 percentage points — option D’s 20%.** **Convention note:** Subtracting two percentages gives a difference in percentage points. A relative percentage change would require a specified denominator and is a different calculation. A: **Incorrect:** 10 percentage points is the 2023 gap, not the maximum. B: **Incorrect:** 40% is a category share in 2021, not the maximum gap. C: **Incorrect:** 50% is Drama’s 2022 share, not the difference between the two shares. D: **Correct:** It corresponds to the maximum 20-percentage-point gap.