Easy -1.8 This is a straightforward calculation requiring only basic formulas for mean and variance from a simple two-value frequency distribution. No problem-solving or conceptual understanding is needed—just direct substitution into standard formulas with minimal arithmetic.
1 A computer can generate random numbers which are either 0 or 2 . On a particular occasion, it generates a set of numbers which consists of 23 zeros and 17 twos. Find the mean and variance of this set of 40 numbers.
For reasonable attempt at the mean using freqs or probs but not using prob=0.5
A1
For correct mean
M1
For correct variance formula
A1ft
For correct answer
4 marks total
Part (ii):
frequencies: 3, 7, 6, 3, 1
scaled frequencies: 3, 7, 3, 1.5, 0.5
Answer
Marks
Guidance
or 0.006, 0.014, 0.006, 0.003, 0.001
M1
For frequencies and attempt at scaling, accept cw/freq but not cw \(\times\) freq, not cw/mid point
Histogram with:
- Correct heights from scaled frequencies
- Correct widths of bars, uniform horiz scale, no halves or gaps or less-than-or-equal tos
Answer
Marks
Guidance
- Both axes labelled, fd and area or m². Not class width
A1
For correct heights from their scaled frequencies seen on the graph
B1
For correct widths of bars, uniform horiz scale, no halves or gaps or less-than-or-equal tos
B1
Both axes labelled, fd and area or m². Not class width
4 marks total
**Part (i):**
$x$ | 0 | 2
freq | 23 | 17
OR
$P(0) = \frac{23}{40}$, $P(2) = \frac{17}{40}$
Mean $= \frac{34}{40} = 0.850$
Variance $= \frac{(4 \times 17)}{40} - (0.85)^2 = 0.978$ (exact answer 0.9775) ($\frac{391}{400}$) | M1 | For reasonable attempt at the mean using freqs or probs but not using prob=0.5
A1 | For correct mean
M1 | For correct variance formula
A1ft | For correct answer
**4 marks total**
**Part (ii):**
frequencies: 3, 7, 6, 3, 1
scaled frequencies: 3, 7, 3, 1.5, 0.5
or 0.006, 0.014, 0.006, 0.003, 0.001 | M1 | For frequencies and attempt at scaling, accept cw/freq but not cw $\times$ freq, not cw/mid point
**Histogram with:**
- Correct heights from scaled frequencies
- Correct widths of bars, uniform horiz scale, no halves or gaps or less-than-or-equal tos
- Both axes labelled, fd and area or m². Not class width | A1 | For correct heights from their scaled frequencies seen on the graph
B1 | For correct widths of bars, uniform horiz scale, no halves or gaps or less-than-or-equal tos
B1 | Both axes labelled, fd and area or m². Not class width
**4 marks total**
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1 A computer can generate random numbers which are either 0 or 2 . On a particular occasion, it generates a set of numbers which consists of 23 zeros and 17 twos. Find the mean and variance of this set of 40 numbers.
\hfill \mbox{\textit{CAIE S1 2003 Q1 [4]}}