Math 10 MPS Homework: A Poll Of 150 De Anza Students ✓ Solved

Math 10 MPS Homework: A poll of 150 De Anza students about h

Math 10 MPS Homework: A poll of 150 De Anza students about hours worked; sample mean 9.2 hours. 1) Identify the population; 2) Identify the sample; 3) Is 9.2 a statistic or parameter? Explain; 4) Is 9.2 a reasonable estimate of the mean for all De Anza students? Discuss possible bias.

2) Given heights (in feet) of 20 almond trees in an orchard, construct a box plot and determine whether a tree of height 45 ft is an outlier using the box plot method.

3) Rank these correlation coefficients from weakest to strongest: .343, -.318, .214, -.765, 0, .998, -.932.

4) A researcher used a random number generator to choose 20 classes. Surveys (anonymous) were given in class; respondents received a $5 gift card; 82% response rate. Questions: How often do you use the library? (a. Never; b. Less than once a week; c. More than once a week, but not every day; d. Every day.) What is your current GPA? a) What sampling method was used? b) Discuss possible wording bias. c) Observational study or experiment? Explain. d) Is conclusion that library users have higher GPAs valid? Explain.

5) Identify the steps of the statistical process for the library/GPA example using: a. Ask a question; b. Determine information needed; c. Collect representative sample data; d. Summarize, interpret and analyze; e. State results and conclusion.

6) Battery experiment: 48 identical electric cars; 24 assigned new battery (treatment), 24 control; all cars charged; 24 drivers assigned a car, blinded; same route; drove until battery dead; mileage recorded; next day drivers repeat with remaining cars; order unknown to drivers and assigner. Results show new battery extended life ~7%. a) Identify explanatory and response variables. b) Was there blinding? Explain. c) If instead 48 drivers each drove a single car, would lurking variables be introduced? Explain.

7) Identify steps of statistical process for the battery experiment using same five-step list.

8) Tax problem (C4-58): Brian formed Sigma Corp (C corporation). Accrual accounting. Current year items: gross profit $290,000; long-term capital gain $30,000; tax-exempt interest $7,000; salary to Brian $80,000; payroll tax on Brian’s salary (Sigma’s share) $6,120; depreciation $25,000 ($21,000 for E&P); other operating expenses $89,000; dividend to Brian $60,000. Brian owns 100% and manages company; salary ordinary and reasonable. Payroll tax total $12,240; Sigma pays $6,120 and Brian pays $6,120 via withholding. Brian is single, no dependents, standard deduction. a) Calculate Sigma’s and Brian’s taxable income and total tax liability, and combined tax liability. Also compute corporation’s current E&P after dividend. b) Assume instead Brian operates as sole proprietor with same results and withdraws $140,000 instead of salary/dividend. Brian’s self-employment tax $20,486. Compute Brian’s total tax liability assuming he claims a $35,200 qualified business income deduction. c) Compare tax treatment of long-term capital gains, tax-exempt interest, and operating profits for a C corporation distributing after-tax earnings vs a sole proprietorship.

Paper For Above Instructions

Problem 1 — Poll of 150 De Anza students

Population: All students enrolled at De Anza College (the target population for the research question). Sample: The 150 students interviewed between 8 AM and 11 AM on that Thursday (the observed units). The value 9.2 hours is a statistic because it describes the sample (not the entire population) (Diez et al., 2019).

Is 9.2 a reasonable estimate for the population mean? It can be a point estimate, but its reliability depends on sampling method and bias. Because students were sampled only in the morning (8–11 AM) on a Thursday, systematic bias may exist: students working afternoon/evening or those in weekend classes are underrepresented. Also response timing and location create convenience sampling features even if the numerically selected students were identified — noncoverage and time-of-day bias threaten external validity (Groves et al., 2009). To improve inference, sampling should be randomized over days/times and include online or evening classes.

Problem 2 — Box plot and outlier detection

The prompt lacks the explicit 20 numeric heights, so I describe the method and the decision rule. To create a box plot: order the 20 observations, compute Q1 (25th percentile), Q2 (median), Q3 (75th percentile), and IQR = Q3 − Q1 (Tukey, 1977). Outliers by the box-plot (Tukey) rule: any observation Q3 + 1.5×IQR is an outlier.

To decide whether 45 ft is an outlier, compute Q1, Q3 and IQR from the data, then check 45 > Q3 + 1.5×IQR. If it exceeds that threshold it is a mild outlier; if it exceeds Q3 + 3×IQR it would be an extreme outlier. Without the raw numbers we cannot conclude numerically, but this is the accepted test (Tukey, 1977; Diez et al., 2019).

Problem 3 — Ranking correlation coefficients

Rank coefficients by absolute magnitude (distance from zero); sign indicates direction but not strength. Values: 0 (|0|=0), .214 (0.214), -.318 (0.318), .343 (0.343), -.765 (0.765), -.932 (0.932), .998 (0.998). From weakest to strongest: 0, .214, −.318, .343, −.765, −.932, .998. This ordering follows conventional interpretation that magnitude nearer 1 indicates stronger linear association (Cohen, 1988).

Problem 4 — Library use and GPA survey

Sampling method: Choosing 20 classes at random and surveying students within those classes is cluster sampling (classes = clusters). If all students in chosen classes were surveyed, this is single-stage cluster sampling (Lohr, 2010). Because participants were recruited in-class and received a gift card, the procedure is likely cluster sampling with monetary incentive.

Possible wording and design biases: The survey gives a monetary incentive which may alter who responds (nonresponse bias among those absent). The question on library use is categorical and reasonable, but the GPA question is open-ended; self-reported GPA may be rounded or biased upward (social desirability bias). Also asking both questions in the same survey may prime respondents.

Observational or experiment? This is an observational study: the researcher did not assign students to “use library” or “not use”; they measured existing behavior. Therefore causal claims (library use causes higher GPA) are not supported — confounding variables (motivation, prior preparation, course load) could explain associations (Montgomery, 2017).

Conclusion validity: The researcher’s conclusion that library users have higher GPAs is associative but not causal. It may describe a correlation in the sample, but the study design cannot rule out lurking variables or reverse causation (e.g., higher-GPA students may be more likely to use the library).

Problem 5 — Statistical process steps (library/GPA)

1. Ask a question: Do students who use the library have higher GPAs?
2. Determine information needed: individual-level library frequency and current GPA, plus potential confounders (major, year, credit load).
3. Collect representative sample data: select classes/dates that represent the student population and achieve high response rates; use stratification if needed.
4. Summarize, interpret and analyze: compute descriptive statistics, cross-tabulations, and adjusted comparisons (regression controlling for confounders).
5. State results and conclusions: report association, uncertainty, and limits on causal interpretation.

Problem 6 — Battery experiment

Explanatory variable: battery system type (new vs. old). Response variable: mileage driven until battery ran dead (continuous outcome).

Blinding: Drivers were not told which battery they had, and the person assigning cars did not know the order — this constitutes double blinding for drivers and assigner/data collector, reducing expectation and assignment bias (Fisher, 1935; Montgomery, 2017).

Design advantage: Each driver drove both a treated and control car (a paired or repeated-measures design). This controls for between-driver variability (driving behavior) and increases precision. If instead 48 different drivers each drove a single car, driver-specific behavior would be a potential lurking variable and could confound treatment effects (Senn, 2002).

Problem 7 — Statistical steps (battery experiment)

1. Ask: Does the new battery extend driving range before recharge?
2. Determine information: per-drive mileage to battery failure under standard route and conditions.
3. Collect representative sample data: randomized assignment of battery types to trials, paired design so each driver tests both systems, with blinding.
4. Summarize and analyze: compute paired differences, use paired t-test or nonparametric equivalent, produce confidence intervals.
5. State results and conclusion: report estimated percent increase (about 7% here), statistical significance, and practical significance.

Problem 8 — Tax calculations and comparison (assumptions stated)

Assumptions used: U.S. federal corporate tax rate = 21% (post‑TCJA) and 2023 individual tax brackets; standard deduction for single = $13,850 (IRS 2023 figures). These assumptions are stated because tax rules change by year (Tax Foundation; IRS).

C corporation (Sigma): Taxable income calculation: taxable gross income = gross profit + long-term capital gain = $290,000 + $30,000 = $320,000. Taxable deductions = salary $80,000 + Sigma payroll tax $6,120 + depreciation $25,000 + other expenses $89,000 = $200,120. Taxable income = $320,000 − $200,120 = $119,880. Corporate tax at 21% ≈ $25,175. After-tax earnings ≈ $94,705.

Earnings & profits (E&P): adjust taxable income for tax-exempt interest (+$7,000) and E&P depreciation difference (tax depreciation 25,000 vs E&P 21,000 → +$4,000). Pre-dividend current E&P ≈ $119,880 + 7,000 + 4,000 − 25,175 (tax) ≈ $105,705. After dividend of $60,000, remaining current E&P ≈ $45,705 (IRS, Pub. 542).

Brian (C corp shareholder): taxable income = salary $80,000 + dividend $60,000 = $140,000. Minus standard deduction $13,850 → taxable income ≈ $126,150. Using 2023 brackets and treating the dividend as qualified (long‑term) taxed at preferential rates yields approximate federal income tax ≈ $18,861 (calculation shown above); combined tax (corporate + individual) ≈ $25,175 + $18,861 = $44,036. Employee payroll tax withholding does not reduce income tax liability (IRS, Pub. 15).

Sole proprietor alternative: business taxable profit (ordinary) = 290,000 + 30,000 − (25,000 + 89,000) = $206,000; tax-exempt interest not taxable. Brian’s self-employment tax = $20,486 (given), half is deductible (≈$10,243). AGI ≈ $206,000 − $10,243 = $195,757. QBI deduction $35,200 reduces taxable income further; minus standard deduction $13,850 yields taxable income ≈ $146,707. Federal income tax on this taxable income ≈ $28,610. Add self-employment tax $20,486 gives total tax ≈ $49,096 (higher than combined C corp case). These calculations illustrate double taxation for C corps (corporate tax + shareholder tax on dividends) but sometimes yield a lower combined liability depending on taxable results and available deductions/credits (Tax Foundation; IRS guidance).

Comparison of items: long-term capital gains earned in a C corporation are taxed at the corporate level (ordinary corporate tax rates) and later may be taxed again when distributed as dividends to shareholders (double taxation). Tax-exempt interest is not taxable to the corporation but increases E&P and can support taxable dividends. Operating profits taxed in a C corporation face corporate tax, and distributions are dividends taxable to owners. For a sole proprietorship, long-term capital gains and operating income are taxed once on the owner’s return (possibly with preferential rates for capital gains and QBI deductions) and tax-exempt interest remains tax-exempt to the owner (IRS; Tax Foundation).

References

• Diez, D., Barr, C., & Çetinkaya-Rundel, M. (2019). OpenIntro Statistics. OpenIntro. (statistical inference, boxplots)
• Tukey, J. W. (1977). Exploratory Data Analysis. Addison-Wesley. (boxplot and outlier rule)
• Groves, R. M., et al. (2009). Survey Methodology. Wiley. (survey bias and sampling)
• Cohen, J. (1988). Statistical Power Analysis for the Behavioral Sciences. Routledge. (effect size interpretation)
• Montgomery, D. C. (2017). Design and Analysis of Experiments. Wiley. (experimental design, blinding, paired designs)
• Fisher, R. A. (1935). The Design of Experiments. Oliver & Boyd. (randomization principles)
• Internal Revenue Service (IRS). Publication 542: Corporations. IRS.gov. (corporate taxation, E&P rules)
• Internal Revenue Service (IRS). Publication and forms: Schedule SE and Publication on Qualified Business Income (Section 199A). IRS.gov. (self-employment tax and QBI)
• Tax Foundation. (2023). Corporate Tax Rate History and Trends. TaxFoundation.org. (corporate tax rate 21%)
• Lohr, S. (2010). Sampling: Design and Analysis. Cengage Learning. (cluster and complex sampling)