Using Only The 'age' Column For The Dataset, Complete The Fo ✓ Solved

Using only the 'age' column for the dataset, complete the fo

Using only the 'age' column for the dataset, complete the following: 1. Create an age frequency distribution: choose class width, list class limits, midpoints, frequencies, relative frequencies, cumulative relative frequencies, and construct an ogive and a frequency polygon. 2. Calculate descriptive statistics: mean (round to two decimals), median (one decimal), sample standard deviation (two decimals), Q1 and Q3 (one decimal). 3. Construct a 95% confidence interval for the average age: report sample mean, sample standard deviation, sample size, choose appropriate distribution (z or t), critical value (two decimals), margin of error (two decimals), lower and upper bounds (two decimals), and interpret. 4. Construct a 95% confidence interval for the proportion of male students: report sample size, number of males, male and female proportions (four decimals), distribution, critical value (two decimals), margin of error (four decimals), lower and upper bounds (four decimals), and interpret. 5. Conduct hypothesis tests at alpha=0.05: a. Test the claim that the average age is 32 years: state H0 and Ha, indicate distribution, compute sample mean, sample standard deviation, test statistic (two decimals), p-value (four decimals), and interpret. b. Test the claim that the proportion of males is 35%: state H0 and Ha, report sample proportions, distribution, test statistic (two decimals), p-value (four decimals), and interpret.

Paper For Above Instructions

Overview

This report describes the procedures and calculations required to analyze an "age" column from an online-student dataset. The analysis covers frequency distributions and graphical displays, descriptive statistics (mean, median, standard deviation, quartiles), two 95% confidence intervals (one for the mean age and one for the male proportion), and two hypothesis tests at alpha = 0.05 (mean age equals 32 and male proportion equals 0.35). Where numeric computation is required, I provide formulas, stepwise methods, and a worked hypothetical example to illustrate each step. The methods follow standard statistical practice (Wackerly et al., 2008; Agresti & Franklin, 2017).

1. Frequency Distribution, Ogive, and Polygon

Procedure:

  1. Decide on the number of classes (k). Common choices use Sturges' rule: k ≈ 1 + log2(n), where n is sample size (Moore et al., 2018).
  2. Compute range = max(age) − min(age). Choose class width = range / k, then round to a convenient value (e.g., 2, 5, or 10).
  3. Define class limits as consecutive non-overlapping intervals covering all ages. Compute midpoint for each class = (lower + upper)/2.
  4. Tally frequencies by counting how many ages fall in each class. Compute relative frequency = frequency / n, and cumulative relative frequency by summing relative frequencies from the lowest class upward.
  5. Plot the ogive (cumulative relative frequency vs. upper class boundary) and the frequency polygon (midpoints on x-axis, frequency on y-axis joined by straight lines) (McClave & Sincich, 2017).

Illustrative example (n = 30, ages hypothetically between 18 and 62): if k = 6 and range = 44, width ≈ 7.3 → choose width = 8. Classes: 18–25, 26–33, 34–41, 42–49, 50–57, 58–65. Compute midpoints (21.5, 29.5, etc.), frequencies from the data, relative frequencies = freq/30, and cumulative relative frequencies. Use those to draw ogive and polygon (graphing software or Excel is recommended) (NIST/SEMATECH, 2012).

2. Descriptive Statistics

Formulas and steps:

  • Mean (sample): x̄ = (Σxi)/n. Round to two decimals.
  • Median: sort ages; if n odd, median is middle value; if even, median is average of two middle values. Round to one decimal.
  • Sample standard deviation: s = sqrt[ Σ(xi − x̄)² / (n − 1) ]. Round to two decimals (Wackerly et al., 2008).
  • Quartiles Q1 and Q3: use the median-splitting method or interpolation (software often uses linear interpolation); round to one decimal.

Example: with hypothetical ages producing x̄ = 35.47, median = 34.0, s = 9.12, Q1 = 27.0, Q3 = 43.5 (numbers illustrative). Always report the rounding requested and confirm consistency with the raw age values (Field, 2013).

3. 95% Confidence Interval for the Mean Age

Decision on distribution:

  • If sample size n ≥ 30 and the age distribution is approximately normal or not severely skewed, the t-distribution is still preferred unless population standard deviation is known (t is conservative) (Hogg & Tanis, 2010).
  • Typically use t with df = n − 1 for unknown population σ (Agresti & Franklin, 2017).

Formula: CI = x̄ ± t × (s/√n), where t is t critical value for 95% and df = n − 1. Steps:

  1. Compute x̄, s, n.
  2. Find t* (two decimals) from t-table or software for 0.025 in each tail.
  3. Compute margin of error ME = t* × (s/√n) and round to two decimals.
  4. Lower = x̄ − ME; Upper = x̄ + ME (both to two decimals).
  5. Interpretation: “We are 95% confident that the true mean age of online students lies between Lower and Upper.”

Illustrative calculation: with x̄ = 35.47, s = 9.12, n = 30 → df = 29, t* ≈ 2.045 (two decimals). ME = 2.05 × (9.12/√30) ≈ 3.41 → CI ≈ (32.06, 38.88) (values rounded) (Moore et al., 2018).

4. 95% Confidence Interval for Proportion of Males

When constructing a CI for a proportion p̂ = x/n, use the normal approximation (z) if np̂ and n(1−p̂) ≥ 5; otherwise use exact methods (Agresti & Coull or Wilson) (Agresti & Franklin, 2017).

Standard formula (Wald): CI = p̂ ± z × sqrt[p̂(1−p̂)/n], with z = 1.96 for 95% (two decimals 1.96). Steps:

  1. Compute sample size n and number of males x, then p̂ = x/n. Report p̂ and female proportion (1−p̂) to four decimals.
  2. Check np̂ and n(1−p̂) ≥ 5; if satisfied, compute ME = z* × sqrt[p̂(1−p̂)/n] and round ME to four decimals.
  3. Lower = p̂ − ME; Upper = p̂ + ME (round to four decimals) and interpret in context.

Illustrative example: n = 30, x = 14 → p̂ = 0.4667 (four decimals 0.4667). ME = 1.96 × sqrt(0.4667×0.5333/30) ≈ 0.1748 → CI ≈ (0.2919, 0.6415). Interpretation: 95% confidence that true male proportion is between ~29.19% and ~64.15% (Agresti & Franklin, 2017). For small samples prefer Wilson interval (Brown et al., 2001).

5. Hypothesis Tests (α = 0.05)

a. Test for Mean Age = 32 years

Set hypotheses: H0: μ = 32 versus Ha: μ ≠ 32 (two-tailed). Use t-test with df = n − 1 if σ unknown (common case). Test statistic: t = (x̄ − 32) / (s/√n). Calculate t to two decimals. Compute p-value = 2 × P(Tdf ≥ |t|) and round to four decimals. Decision: if p ≤ 0.05 reject H0; otherwise fail to reject. Interpret in plain language (e.g., insufficient evidence to conclude mean ≠ 32) (Hogg & Tanis, 2010).

Example: x̄ = 35.47, s = 9.12, n = 30 → t = (35.47 − 32)/ (9.12/√30) ≈ 1.99 → p ≈ 0.0550 (two-tailed). Since p > 0.05, fail to reject H0; evidence is not strong enough at 5% level.

b. Test for Proportion of Males = 0.35

Set hypotheses: H0: p = 0.35 versus Ha: p ≠ 0.35 (two-tailed). Use z-test for proportion if np0 and n(1−p0) ≥ 5. Test statistic: z = (p̂ − p0) / sqrt[p0(1−p0)/n]. Compute z to two decimals and p-value = 2 × P(Z ≥ |z|) (round four decimals). Decision rule: reject H0 if p ≤ 0.05.

Example: n = 30, x = 14 → p̂ = 0.4667. Standard error = sqrt(0.35×0.65/30) = 0.0874. z = (0.4667 − 0.35)/0.0874 ≈ 1.34 → p ≈ 0.1790. Fail to reject H0; the observed proportion is not significantly different from 35% at α = 0.05 (Agresti & Franklin, 2017).

Summary and Reporting Recommendations

Follow these reporting conventions:

  • Report sample sizes and exact statistics (x̄, s, p̂) with the requested rounding.
  • Show critical values and test statistics with the requested decimal precision and cite the distribution used (t or z).
  • State decisions and interpret results in context, avoiding technical jargon—e.g., “insufficient evidence” or “statistically significant.”

When producing the deliverables, include the constructed frequency table, ogive, and polygon as images or embedded plots; include formula derivations and the numeric steps used to get each rounded value; and attach the raw age list as an appendix for reproducibility (Field, 2013; NIST/SEMATECH, 2012).

References

  • Agresti, A., & Franklin, C. (2017). Statistical Methods for the Social Sciences (5th ed.). Pearson.
  • Agresti, A., & Coull, B. A. (1998). Approximate is better than “exact” for interval estimation of binomial proportions. The American Statistician, 52(2), 119–126.
  • Brown, L. D., Cai, T. T., & DasGupta, A. (2001). Interval estimation for a binomial proportion. Statistical Science, 16(2), 101–133.
  • Field, A. (2013). Discovering Statistics Using IBM SPSS Statistics (4th ed.). Sage Publications.
  • Hogg, R. V., & Tanis, E. A. (2010). Probability and Statistical Inference (9th ed.). Pearson.
  • McClave, J. T., & Sincich, T. (2017). Statistics (13th ed.). Pearson.
  • Moore, D. S., McCabe, G. P., & Craig, B. A. (2018). Introduction to the Practice of Statistics (9th ed.). W. H. Freeman.
  • NIST/SEMATECH. (2012). e-Handbook of Statistical Methods. National Institute of Standards and Technology. https://www.itl.nist.gov/div898/handbook/
  • Wackerly, D., Mendenhall, W., & Scheaffer, R. (2008). Mathematical Statistics with Applications (7th ed.). Cengage Learning.
  • Wilcox, R. R. (2012). Introduction to Robust Estimation and Hypothesis Testing (3rd ed.). Academic Press.

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