Statistics lab analysis

E1: from values to summaries

Compute mean as sum divided by count. For sample SD, subtract the mean from each value, square each deviation, and add the squares. Divide by n minus one, then take the square root. NIST gives this sample formula.

Mean and SD retain bpm units here. Variance has squared bpm units. Report count for each condition. With paired data, also count complete pairs; do not count two measurements on one person as two independent people.

Overlap of condition values is a useful first look, but it does not determine a paired test. That test uses the matched differences and their spread.

E2: choose the model before the test

NIST describes paired comparisons through one-to-one differences. The paired t statistic is mean difference divided by its standard error. Here, t = mean difference / (SD of differences / square root of n), with df = n minus one.

For a small paired sample, the model needs independent pairs and a reasonably normal population of differences. OpenStax also identifies random sampling. A few tidy values cannot prove these assumptions. They are stipulated for D1 arithmetic only.

For actual work, verify pairing, independent units, sampling, timing, missing values and outliers with the teacher. Independent groups need a different test; do not force them into D1’s paired formula. Never rearrange one column alone.

For D1 choose a two-sided paired test before looking at the result. Null: the modeled population mean B-minus-A difference is zero. Alternative: it is not zero. Set alpha to 0.05.

D1: original recovery data and design

Invented instructional pulse values, not participant observations. Six fictional people each have one A and one B value. A and B are neutral labels; no intervention, collection method or actual outcome is asserted. No data collection is authorized.

Enter this complete table in a blank approved spreadsheet. Row 1 headers: A1 Pair; B1 A (bpm); C1 B (bpm); D1 B minus A (bpm). Rows 2 through 7 are: P1,60,62; P2,62,65; P3,64,65; P4,66,70; P5,68,70; P6,70,73. D cells start blank.

For this practice, assume six independent pairs sampled from a model with normally distributed paired differences. This is a teaching assumption, not a tested claim. All twelve values are present; do not exclude any or add zeros. There are six pairs, not twelve people.

Keep each A/B pair on the same row. Condition A ranges from 60 to 70 bpm; B ranges from 62 to 73 bpm. The intervals overlap. Compute both means by hand before using functions.

R1: reproducible spreadsheet steps

Create a sheet named Practice with D1’s headers and rows. Label the workbook Original recovery data. Compare every entered value against D1 before calculating. Use the following Excel formulas; another program needs its verified equivalent.

Enter =C2-B2 in D2 and fill through D7. In F1:I1 enter Summary, A, B, Difference. Put Mean, Sample SD and Count in F2:F4. In G2 enter =AVERAGE(B2:B7); G3 =STDEV.S(B2:B7); G4 =COUNT(B2:B7).

In H2:H4 use the same functions with C2:C7. In I2:I4 use them with D2:D7. These are three separate formulas per column. Display two decimal places for means and SDs, but keep full stored precision.

In F6 write Paired two-sided p. In G6 enter =T.TEST(C2:C7,B2:B7,2,1). Microsoft defines 2 as two tails and 1 as paired. Display at least six decimal places for p. Do not replace it with zero.

In F7 write Alpha and enter 0.05 in G7. In F8 write Decision and enter your comparison in G8. In F9 write Mean B minus A (bpm) and enter =AVERAGE(D2:D7) in G9.

On a Notes sheet record data origin, sheet/ranges, units, six pairs, hypotheses, assumptions, exclusions (none for D1), formulas, software/version and rounding. Add the numerical result, interpretation and limitations. Save both sheets in the same workbook.

If spreadsheet access is missing, calculate and interpret E4 on paper as preparation. Ask how to complete the spreadsheet execution and file submission; after uploading, check that Schoology shows Submitted. This does not replace the required spreadsheet product.

E3: what a p-value says

A p-value describes compatibility with a specified null model. It is the probability, under that model and its assumptions, of a statistic at least as extreme as the calculated statistic.

For this exercise compare p with alpha = 0.05. A lower p supports rejecting the zero-mean-difference null within the assumed model. A higher p means insufficient evidence to reject it; it does not prove equality.

The ASA warns that a p-value does not give the probability a hypothesis is true or measure effect size. It does not prove causation, practical importance or future replication. Show the difference and limitations alongside it.

Running documented steps on the same data can reproduce a calculation. Replicating a biological result needs new observations. Neither a reproducible mistake nor a low p-value guarantees a true clinical claim.

E4: recovery check results

Check after attempting D1. Each count should be 6. Each mean should equal its column sum divided by 6 and fall inside that column’s range. Recompute one sample SD by hand with E1’s steps.

Subtract A from B on each row, keeping every pair on its own row. The mean difference should equal mean B minus mean A. Six complete pairs give df = 5.

Your two-sided paired p-value comes from G6’s T.TEST formula; check that it uses both columns, 2 tails and type 1, and displays at least six decimal places. Compare it with alpha = 0.05 and state your decision about the zero-mean-difference null under the stipulated model.

Report the mean B-minus-A difference in bpm from your sheet. These invented numbers establish no actual change in a person and no clinical benefit. If your sheet differs, check ranges, paired rows, sample SD and test arguments before changing data.

P1: actual work and official holds

Your actual-data analysis remains required when assigned. Recover the submitted physiology table and March 11 graph locally. Check their match, units, conditions, pair structure, independent units and missing values. Do not paste private records into public work.

Get the teacher’s approved test, direction, threshold and missing-value policy before applying functions to actual data. Document your real sheet names and ranges. Ask the teacher to resolve any unknown design decision; do not copy D1 assumptions onto a different design.

The absence route uses D1 and R1 as original shared practice. Follow the written steps and run the test when software is available. Then identify what is needed to bring your own data through the appropriate path. D1 is not the missing teacher-provided dataset or your collected observations.

The async task remains spreadsheet summaries, a run or interpretation of a t-test and reproducible steps. No homework is due before this class. Actual spreadsheet access and assigned data are still necessary.

Your teacher may assign Activity 2.1.3 Making Results Meaningful and will confirm its screens, format and stop point. An original workbook does not mark that official activity or tracker complete.

Weekly physiology table, sensor lab, graphing, mean/SD/error and Problem 2 physiology-data evidence remain separate. The weekly supplied biometric dataset is not provided here. Unit bias/error, graph, data-based CER and replication limitation also remain open for actual evidence. Keep work for March 16’s CER mini-report.

X1: warmups, checks and sample size

Warmup one: two means differ by 3. What else is needed? Name units, spreads, counts and design or pairing. Warmup two: explain why another reader needs exact data and steps to reproduce a calculation.

Self-check is hypothetical: p = 0.02 with alpha = 0.05. That crosses the cutoff. Under the null and assumptions, a statistic at least this extreme has probability 0.02. It does not mean a 98 percent chance that the effect is real.

Extension: in two sentences explain p = 0.03 and what it does not mean. Then name why a small sample could miss a real effect. Its uncertainty can be too large relative to the effect to cross the cutoff.

Exit: doubling independent sample size can reduce standard error if spread and design stay comparable. It does not guarantee a changed decision because new values and assumptions matter. Copying existing rows doubles a row count, not independent information.

Restart at the first unfinished step: summaries, appropriate test or teacher-confirmed result interpretation, decision, then reproducible notes. Keep actual and recovery results clearly labeled.

Verify inputs and estimate by hand

Choose actual data with confirmed design or labeled D1 recovery. Compute both condition means by hand and describe overlap. Answer both X1 warmups.

D1 supplies all recovery inputs. Overlap alone does not determine the paired result.

Spreadsheet summary table

Use R1 to calculate mean, sample SD and count per condition, plus paired differences where appropriate. Record units and complete-pair count.

For D1 use six rows and sample SD. Compare with E4 after trying. Do not use invented values for missing real observations.

Comparison test and interpretation

State hypotheses, design, assumptions, alpha and test type. Run the appropriate test or interpret the D1 result your teacher confirms. Report p, mean difference and a bounded conclusion.

R1/E2/E3/E4 supply the recovery method. Ask for the missing actual design or software. No causal or clinical claim follows.

Reproducible analysis record

Document data origin, ranges, units, formulas, software, exclusions, precision and interpretation on Notes. Submit the same workbook through the confirmed route.

Keep Practice and Notes together. State whether this is recovery or actual analysis, and list remaining official/data tasks.

Significance and doubled-sample check

Answer X1’s p = 0.02 check and doubled-sample exit. If ready, add the p = 0.03 extension and a small-sample missed-effect explanation.

State the null-model condition. Neither p nor more rows proves a result real, important or repeatable.