SPS SPS SM Statistics (SPS SM Statistics) 2025 April

1. It is known that, under standard conditions, \(12 \%\) of light bulbs from a certain manufacturer have a defect. A quality improvement process has been implemented, and a random sample of 200 light bulbs produced after the improvements was selected. It was found that 15 of the 200 light bulbs were defective.
   1. State one assumption needed in order to use a binomial model for the number of defective light bulbs in the sample.
   2. Test, at the \(5 \%\) significance level, whether the proportion of defective light bulbs has decreased under the new process.
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   The histogram shows information about the lengths, \(l\) centimetres, of a sample of worms of a certain species.
   \includegraphics[max width=\textwidth, alt={}, center]{a18b06b1-053e-45b2-9c28-2f125cf6cbba-06_805_1151_269_280} The number of worms in the sample with lengths in the class \(3 \leqslant l < 4\) is 30 .
2. Find the number of worms in the sample with lengths in the class \(0 \leqslant l < 2\).
3. Find an estimate of the number of worms in the sample with lengths in the range \(4.5 \leqslant l < 5.5\).
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3. A researcher has collected data on the heights of a sample of adults but has encoded the actual values using a linear transformation of the form \(a X + b\), where \(X\) represents the original height in centimetres.
Given the following information about the encoded data:
The mean of the encoded heights is 5.4 cm
The standard deviation of the encoded heights is 2.0 cm
The researcher knows that the transformation used was \(0.2 X - 30\)

1. Find the mean of the original heights in the sample.
2. Find the standard deviation of the original heights in the sample.
3. If an encoded height value is 6.8 , what was the original height in centimetres?
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4. A manufacturing plant produces electronic circuit boards that need to pass two quality checks - a mechanical inspection and an electrical test. Historical data shows that \(15 \%\) of boards fail the mechanical inspection. Of those that pass the mechanical inspection, \(8 \%\) fail the electrical test. Of those that fail the mechanical inspection, \(60 \%\) fail the electrical test.

1. If a board is randomly selected from production, what is the probability that it passes both inspections?
2. If a board is selected at random and is found to have passed the electrical test, what is the probability that it also passed the mechanical inspection?
3. The company continues to test boards from a large batch until finding one that passes both inspections. Each board is tested independently of all others. What is the probability that they need to test exactly 3 boards to find one that passes both inspections?
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5. In a study of reaction times, 25 participants completed a test where their reaction times (in milliseconds) were recorded. The results are shown in the stem-and-leaf diagram below: \(20 \mid 3579\)
\(21 \mid 02568\)
\(22 \mid 134579\)
\(23 \mid 0258\)
\(24 \mid 1467\)
\(25 \mid 25\) Key: 21 | 0 represents a reaction time of 210 milliseconds

1. State the median reaction time.
2. Calculate the interquartile range of these reaction times.
3. Find the mean and standard deviation of these reaction times.
4. State one advantage of using a stem-and-leaf diagram to display this data rather than a frequency table.
5. One participant completed the test again and recorded a reaction time of 195 milliseconds. Add this result to the stem-and-leaf diagram and state the effect this would have on:
   a. the median
   b. the mean
   c. the standard deviation
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6. Explain why the interquartile range might be preferred to the standard deviation as a measure of spread in this context
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6. A retail bakery makes cherry muffins where, due to the production process, \(15 \%\) of muffins contain a lower than expected quantity of cherries. The bakery sells these muffins in boxes of 20.

1. State a suitable distribution to model the number of muffins with a lower than expected quantity of cherries in a box, giving the value(s) of any parameter(s). State any assumptions needed for your model to be valid.
2. Using your model from part (a), find the probability that a randomly selected box contains:
   1. exactly 3 muffins with a lower than expected quantity of cherries,
   2. at least 5 muffins with a lower than expected quantity of cherries.
3. The bakery sells 25 boxes of muffins in one day. Find the probability that fewer than 4 of these boxes contain exactly 3 muffins with a lower than expected quantity of cherries.
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7. Miguel has six numbered tiles, labelled \(2,2,3,3,4,4\). He selects two tiles at random, without replacement. The variable \(M\) denotes the sum of the numbers on the two tiles.

1. Show that \(P ( M = 6 ) = \frac { 1 } { 3 }\) The table shows the probability distribution of \(M\)

   \(m\)	4	5	6	7	8
   \(P ( M = m )\)	\(\frac { 1 } { 15 }\)	\(\frac { 4 } { 15 }\)	\(\frac { 1 } { 3 }\)	\(\frac { 4 } { 15 }\)	\(\frac { 1 } { 15 }\)

   Miguel returns the two tiles to the collection. Now Sofia selects two tiles at random from the six tiles, without replacement. The variable \(S\) denotes the sum of the numbers on the two tiles that Sofia selects.
2. Find \(P ( M = S )\)
3. Find \(P ( S = 7 \mid M = S )\)
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