Question 10
Two types of material (A and B) are being tested for their ability to withstand different temperatures. A random selection of both materials was subjected to extreme temperature changes and then classified according to their condition after they were removed from the testing facility. The results are shown in the table.
| Material | |||
|---|---|---|---|
| A | B | Total | |
| Broke completely | 25 | 43 | 68 |
| Showed defects | 35 | 38 | 73 |
| Remained intact | 35 | 24 | 59 |
| Total | 95 | 105 | 200 |
An approximate 95% confidence interval for the probability that material A will break completely or show defects is given by
$\left(c-1.96\sqrt{\dfrac{c(1-c)}{n}}\right.,$$\left.c+1.96\sqrt{\dfrac{c(1-c)}{n}}\right)$
The values of $c$ and $n$ are
That’s it. Step through the hints below to see why.
Not quite. 60 is the right count, but 200 is every component tested, both materials. The question is about material A only, so divide by material A’s total, 95. That also matches $n=95$.
Not quite. 140 is $200-60$, so $\tfrac{140}{200}=1-\tfrac{60}{200}$. That counts the components that were not material A breaking or showing defects: all 105 of material B plus material A’s 35 intact. It’s the complement, taken over the whole table. (Across both materials, $68+73=141$ broke or showed defects, so 140 isn’t that count either.) Material A had $25+35=60$ that broke completely or showed defects, out of 95.
Not quite. This treats all 200 components as the sample. The interval is for material A, so the sample is the 95 material A components: $c=\tfrac{60}{95}$ and $n=95$.
Each bar runs left to right: broke completely, showed defects, remained intact. The number line shows each option’s interval, worked out from its values of $c$ and $n$.
Material A, broke completely or showed defects: $c=\tfrac{60}{95}$ and $n=95$. Your interval matches option (A).
Choose at least one outcome.
Every outcome is chosen, so $c=1$. Every component did one of the three things. Choose only the outcomes the question names.
Right group, $n=95$. Now choose the outcomes the question names: broke completely or showed defects.
This is material B, $n=105$. The question asks about material A.
This uses both materials, so $n=200$. The question asks about material A only.
Your interval lands exactly on option (A), but only by coincidence. Material A had 35 with defects and 35 intact, so broke or intact also gives $c=\tfrac{60}{95}$. The question names broke completely or showed defects.
Your interval looks close to option ()’s, but it isn’t the same one. Compare the values of $c$ and $n$ in the readout.
In options (B) and (C), the denominator of $c$ doesn’t match $n$. Option (D) is consistent, but its sample is all 200 components, not material A.
The interval is for material A only, so the sample is the material A components. Read the total at the bottom of the A column.
$n=95$
A “success” here is a material A component that broke completely or showed defects. Add those two cells.
$25+35=60$$c=\hat p=\dfrac{60}{95}$
The interval has the form $\hat p\pm1.96\sqrt{\dfrac{\hat p(1-\hat p)}{n}}$, so $c$ is the sample proportion and $n$ is the sample size.
Both must come from material A’s column. The 200 counts both materials, so it doesn’t belong in either.
$c=\hat p=\dfrac{25+35}{95}=\dfrac{60}{95}$$n=95$
Use only material A’s column for both the count and the total.
Only mark this done when you could do it without help. Reading the solution does not count.
Question wording is from the 2020 QCAA Mathematical Methods external assessment, © State of Queensland (QCAA) 2020, licensed under CC BY 4.0, and has been adapted. Tangent Tuition is not affiliated with the QCAA.
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