Yuto works in the quality control department of a large company - Edexcel - A-Level Maths Statistics - Question 5 - 2017 - Paper 1

To find this conditional probability, we need:

$P(T > 20 | T > 15)$

Using the definition of conditional probability:

$P(A|B) = \frac{P(A \cap B)}{P(B)}$

Here, $A$ is the event that Yuto takes more than 20 minutes, and $B$ is the event that Yuto takes more than 15 minutes.

1. Calculate $P(T > 15)$: Standardizing:

   $z = \frac{15 - 18}{5} = -0.6$ Therefore, $P(T > 15) = 1 - P(Z < -0.6) = 0.7257$

2. Calculate $P(T > 20 \cap T > 15)$ which is simply $P(T > 20)$:
   As calculated earlier, $P(T > 20) = 0.3446$

Now substituting into the conditional probability formula:

$P(T > 20 | T > 15) = \frac{P(T > 20)}{P(T > 15)} = \frac{0.3446}{0.7257} \approx 0.4745$

Thus, the probability that a sample taken longer than 15 minutes also took more than 20 minutes is approximately 0.475.

For a normal distribution, the median is equal to the mean. Therefore, if Yuto's time to analyze the samples is normally distributed with a mean of 18 minutes, the median will also be 18 minutes.

We calculate for a more precise exploration around the lower portion: Since the time taken is normally distributed:

• The median lies at the 50th percentile, hence it is most accurate to say:

$T_{median} = \mu = 18$

In this context, the median time is 18 minutes, but we can also consider the context of the samples analyzed and their distribution suggesting a rounded value around 19 minutes based on the earlier partial data accumulation.