Independent study

Keywords

independent study, z-score exercises, empirical rule practice, normality check, self-check

Self-learning Normal distribution practice 6 hours · 5 activities

Acquisition

1 Read and make notes

⏱ 60 min

Read the assigned sections and make notes on quartiles, outliers, the normal curve and z-scores.

Production

2 Guided practice

⏱ 75 min

Complete guided questions on skewness, the empirical rule and standardised scores.

Production

3 Independent analysis

⏱ 75 min

Repeat one normality check and one z-score interpretation task in your own time.

Production

4 Reflection or study note

⏱ 60 min

Write a short note on why a manager should care about unusually high or low values.

Assessment

5 Checkpoint

⏱ 90 min

Complete the chapter checkpoint quiz.

NoteNothing on this page is submitted

This page is for your own study and your preparation for the final examination. It is ungraded, and everything you do here stays with you. The chapter’s graded work is the checkpoint quiz.

You may use Google NotebookLM on this page: to quiz yourself on the exercises, to explain a step a second way, or to turn your notes into a revision sheet. Keep what you produce for your exam revision. NotebookLM can make mistakes, so check each answer against the chapter before you rely on it.

Reading

The assigned reading is this chapter and two sections of chapter 1. Read the explanatory sections of the chapter. At a working pace with a pen, this takes about one hour.

Complete the 10 embedded Try It exercises from the chapter text during Activity 2.

Make your notes under four headings: quartiles and outliers, shape, the normal curve, and z-scores. Each heading should include its definition, its formula where it has one, and one example from the teaching dataset. The chapter’s Key terms and Formula review are a good check: if your notes cover those, you have the chapter.

If a section went too fast: OpenStax, Introductory Business Statistics 2e, is free to read at openstax.org. Chapter 6 covers the normal distribution and z-scores at a slower pace with a second set of examples. This is optional and falls outside the allocated hour.

Guided practice

Complete the 10 embedded Try It exercises from the chapter text before starting the four additional scenarios below. The solutions for the additional scenarios are at the bottom of this page.

Try It A: Read the shape

JASP reports these figures for the number of items in 200 online orders:

Table 2.23: Descriptive statistics for items per order
Statistic Value
Mean 4.8
Median 4.0
Skewness 0.91
SE of skewness 0.172
Excess kurtosis 0.64
SE of kurtosis 0.342
  1. Calculate the skewness ratio and the kurtosis ratio.
  2. Describe the shape in one sentence.
  3. Which measure of centre would you report as the typical order size?

Try It B: The empirical rule and Chebyshev

A bakery’s daily sales of loaves have a mean of 340 and a standard deviation of 25.

  1. If sales are close to normal, between which two figures do sales fall on about 68 per cent of days? On about 95 per cent?
  2. On roughly how many days in a 360-day year would sales exceed 390?
  3. If the shape of the distribution were unknown, what share of days at least would fall between 290 and 390?

Try It C: z-scores in context

Two regional managers each claim the better month. The North region sold 1,240 units, where its monthly mean is 1,100 and its standard deviation is 80. The South region sold 860 units, where its monthly mean is 700 and its standard deviation is 120.

  1. Calculate each region’s z-score for the month.
  2. Which region had the more exceptional month, relative to its own history?
  3. Classify each month as typical, unusual or extreme.

Try It D: Shares and percentiles

Delivery times from a warehouse are close to normal, with a mean of 48 hours and a standard deviation of 6 hours.

  1. What share of deliveries take longer than 60 hours?
  2. What share take between 42 and 57 hours?
  3. The firm wants to promise a delivery time that 95 per cent of orders meet. What should the promise be?

Independent analysis

Apply the tutorial tasks to a new variable: Satisfaction_Post. This variable records each customer’s satisfaction score after the service change on the same 1 to 5 scale as Satisfaction. Use JASP or PocketStat.

  1. One normality check: Run the five normality checks from part 3 on Satisfaction_Post. If you use PocketStat, run the four available checks. Fill in one column of the record table from the tutorial and provide a one-line verdict. Then write the Shapiro-Wilk result twice, once for a manager and once as an APA sentence, following two ways to report a result.
  2. One z-score interpretation: Find the lowest Satisfaction_Post score in the file. Calculate its z-score by hand and classify it. Write two sentences for a manager about that customer in plain language without using statistical symbols.

Keep a clean copy of your output, with a note attached as in the tutorial, for your revision.

Open this only after you have your own output. These are the figures to check against. The interpretation is yours to write.

  • Mean 4.138, median 4.285, standard deviation 0.761, minimum 1.59, maximum 5.00
  • 37 of the 200 customers score exactly 5.00
  • Skewness \(-0.725\) in JASP, standard error 0.172, ratio \(-4.22\)
  • Excess kurtosis \(-0.100\), standard error 0.342
  • Shapiro-Wilk \(W = .920\), \(p < .001\)
  • The lowest score is respondent 190’s, 1.59, with a z-score of \(-3.35\)

If your figures match, check your verdict against the ceiling effect in part 1. If they differ, check the variable type first.

Reflection or study note

Write a study note of 250 to 300 words answering one question: why should a manager care about unusually high or low values?

A strong note includes four elements:

  1. Give one example of an unusual value that signals a problem, and one that signals an opportunity. The teaching dataset contains both: the 90-minute wait and the customers who spent more than 1,000 USD.
  2. Explain the difference between flagging a value and deleting it.
  3. State when the z-score rule suits the data and when the IQR fences from chapter 1 suit it better.
  4. Name one decision in a workplace you know where the tails matter more than the average.

Self-check

Use this list to check your own work before the checkpoint quiz.

Table 2.24: Self-check before the checkpoint quiz
Check
☐ My normality record has a verdict for all four tutorial variables
☐ I read skewness and kurtosis against their standard errors
☐ My verdict uses the pictures and the numbers together
☐ Row 1 of my z-score column matches my own calculation, -1.466
☐ I classified each flagged case as unusual or extreme
☐ My annotation names a decision or a warning
☐ My annotation is under 80 words
☐ I used the normal model only on a variable that passed the checks
☐ My APA sentence drops the zero before the decimal point in \(W\) and \(p\)
☐ I have said whether I used JASP or PocketStat

Review questions

  1. Why do two distributions with the same mean and standard deviation present different risks for a manager?
  2. What does a skewness ratio beyond 2 tell you that the skewness alone does not?
  3. State the empirical rule, and name the one condition it needs.
  4. Why does a z-score have no units, and why is that useful?
  5. A Shapiro-Wilk test on 2,000 cases returns \(p = .010\) for a variable whose histogram looks close to normal. What would you conclude?
  6. Why is the normal model a poor choice for a satisfaction score that many customers rate at the maximum?

Solutions

Solutions to the 10 embedded chapter exercises appear directly beneath each task in the chapter text, so you can check an answer without losing your place. The four tasks below are answered here.

  1. Skewness ratio: \(0.91 \div 0.172 = 5.29\). Kurtosis ratio: \(0.64 \div 0.342 = 1.87\).
  2. Order size is moderately skewed to the right, with a ratio well beyond 2, and its tails are close to normal: most orders are small, and a minority of large orders stretch the right tail.
  3. The median, 4 items. The mean of 4.8 is pulled up by the large orders.
  1. About 68 per cent: \(340 \pm 25\), from 315 to 365 loaves. About 95 per cent: \(340 \pm 2 \times 25\), from 290 to 390 loaves.
  2. 390 is two standard deviations above the mean, so about 2.5 per cent of days exceed it: $0.025 = $ 9 days.
  3. 290 to 390 is \(k = 2\), so Chebyshev’s rule guarantees at least $1 - 1/4 = $ 75 per cent of days.
  1. North: \(z = (1{,}240 - 1{,}100) \div 80 = 140 \div 80 = 1.75\). South: \(z = (860 - 700) \div 120 = 160 \div 120 = 1.33\).
  2. North: Its month is 1.75 standard deviations above its own mean, against 1.33 for South, even though South’s rise in units was larger.
  3. Both are typical, since neither reaches 2. North’s month is a strong one, and it falls inside the range that contains about 95 months in 100.
  1. \(z = (60 - 48) \div 6 = 2.0\). Area above 2.0 is 0.0228, about 2.3 per cent.
  2. \(z_{42} = (42 - 48) \div 6 = -1.0\) and \(z_{57} = (57 - 48) \div 6 = 1.5\). Area below \(-1.0\) is 0.1587, and area above 1.5 is 0.0668. Between: $1 - 0.1587 - 0.0668 = $ 0.7745, about 77.5 per cent.
  3. The 95th percentile, \(z = 1.645\): $48 + 1.645 = 48 + 9.9 = $ 57.9 hours. A promise of 58 hours would be met by about 95 per cent of orders.

Key takeaways

  • Look at the shape before you choose a summary or a model.
  • Judge skewness and kurtosis by their size and by their ratio to the standard error.
  • The empirical rule applies to distributions close to normal. Chebyshev’s rule applies to any shape.
  • A z-score places a value in standard deviations from the mean, and it compares values measured in different units.
  • A value beyond 2 is unusual and beyond 3 is extreme. Both call for a second look at the record.
  • Check a variable before you use the normal model, and for skewed or capped variables report percentiles taken from the data.