Problem Details:

The amount of drink dispensed by the machine follows a Normal distribution:

• Mean ($\mu$) = 200 ml
• Variance ($\sigma^2$) = 225 ml$^2$
• Standard Deviation ($\sigma$) = $\sqrt{225} = 15$ ml

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Part (i): Probability that a cup contains between 191 to 209 millilitres

We need to calculate $P(191 \leq X \leq 209)$, where $X \sim N(200, 15^2)$.

1. Standardize the values (using the formula $Z = \frac{X - \mu}{\sigma}$): $Z_{191} = \frac{191 - 200}{15} = -0.6$ $Z_{209} = \frac{209 - 200}{15} = 0.6$

2. Use the standard normal table (or a calculator) to find the probabilities:

   • $P(Z \leq 0.6) \approx 0.7257$
   • $P(Z \leq -0.6) \approx 0.2743$

3. The probability between $Z = -0.6$ and $Z = 0.6$: $P(-0.6 \leq Z \leq 0.6) = P(Z \leq 0.6) - P(Z \leq -0.6)$ $P(-0.6 \leq Z \leq 0.6) = 0.7257 - 0.2743 = 0.4514$

Thus, the probability is approximately 0.4514 or 45.14%.

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Part (ii): Number of cups that will overflow (if 230 ml cups are used)

We need to calculate $P(X > 230)$, where $X \sim N(200, 15^2)$.

1. Standardize the value: $Z_{230} = \frac{230 - 200}{15} = 2$

2. Use the standard normal table to find $P(Z > 2)$:

   • $P(Z > 2) = 1 - P(Z \leq 2)$
   • From the table: $P(Z \leq 2) \approx 0.9772$
   • $P(Z > 2) = 1 - 0.9772 = 0.0228$

3. For 1000 drinks, the expected number of overflows: $\text{Expected overflows} = 0.0228 \times 1000 = 22.8$

Thus, approximately 23 cups will overflow.

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Part (iii): Smallest 25% of the drinks

We need to find the value of $X$ such that $P(X \leq k) = 0.25$, where $X \sim N(200, 15^2)$.

1. Use the standard normal table to find the $Z$-value corresponding to $P(Z \leq -0.674) = 0.25$.

2. Convert $Z$ back to the original $X$-scale: $X = Z \cdot \sigma + \mu$ $X = -0.674 \cdot 15 + 200 = 200 - 10.11 = 189.89$

Thus, the smallest 25% of the drinks are below approximately 189.89 ml.

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Summary of Results:

1. Part (i): Probability = 0.4514 or 45.14%
2. Part (ii): Expected overflows = 23 cups
3. Part (iii): Smallest 25% below 189.89 ml

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Do you want a detailed explanation or graphs for these calculations?

5 Related Questions:

1. What is the probability that a cup contains more than 215 ml?
2. If the variance increases, how would it affect the probabilities calculated above?
3. What is the probability that a cup contains less than 191 ml?
4. How many cups will likely overflow if 220 ml cups are used?
5. How would the calculations change if the mean was adjusted to 210 ml?

Tip:

When solving Normal distribution problems, always sketch the curve and mark the critical areas to visualize probabilities.