A small juice bottling startup wants to monitor the consistency of 3-ounce bottles. Five samples of 3 bottles each were measured (in ounces), what are the control limits for the R-chart?

Sample

Bottle 1

Bottle 2

Bottle 3

1

3

3

3

2

3

3

2

3

2

2

3

4

3

3

3

5

2

3

2

Question

A small juice bottling startup wants to monitor the consistency of 3-ounce bottles. Five samples of 3 bottles each were measured (in ounces), what are the control limits for the R-chart?

Sample

Bottle 1

Bottle 2

Bottle 3

1

3

2

3

2

3

2

3

4

3

5

2

3

2

BlackTom AI · Accepted Answer

We need to determine the R-chart control limits for the given data and compare them to the provided options.
First, compute the range for each subsample (each sample has 3 bottles):
- Sample 1: values 3, 3, 3 → range = max - min = 3 - 3 = 0
- Sample 2: values 3, 3, 2 → range = 3 - 2 = 1
- Sample 3: values 2, 2, 3 → range = 3 - 2 = 1
- Sample 4: values 3, 3, 3 → range = 3 - 3 = 0
- Sample 5: values 2, 3, 2 → range = 3 - 2 = 1
Thus theR-bar, the average range, is (0 + 1 + 1 + 0 + 1) / 5 = 3/5 = 0.6.

Next, obtain the factors for the R-chart given the subgrp size n = 3. For n = 3, the standard control-chart constants are D3 = 0 and D4 ≈ 2.574 (these are the factors used to compute LCL = D3·R-bar and UCL = D4·R-bar).
Therefore:
- LCL = D3 × R-bar = 0 × 0.6 = 0
- UCL = D4 × R-bar ≈ 2.574 × 0.6 ≈ 1.5444

Now compare these calculated limits to the provided options:
- Option A: LCL = 0, UCL = 0.6 → UCL is too low (0.6 ≠ 1.544).
- Option B: LCL = 0, UCL = 1.8 → UCL is higher than our calculation (1.8 ≠ 1.544).
- Option C: LCL = 0, UCL = 1.269 → UCL is lower than our calculation (1.269 ≠ 1.544).
None of the listed options match the standard calculation with R-bar = 0.6 and D4 ≈ 2.574.

Consequence: since none of the provided numeric pairs equal the computed LCL and UCL, the correct response among the choices is effectively that 'Neither is correct'. Each incorrect option misstates either the UCL (too low or too high) or, in one case, retains the correct LCL of 0 but still uses an incorrect UCL value, revealing a mismatch with the standard R-chart constants for n = 3.

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A small juice bottling startup wants to monitor the consistency of 3-ounce bottles. Five samples of 3 bottles each were measured (in ounces), what are the control limits for the R-chart? Sample Bottle 1 Bottle 2 Bottle 3 1 3 3 3 2 3 3 2 3 2 2 3 4 3 3 3 5 2 3 2

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