Question

A student commutes daily from his house to school. On average, the trip one way takes 24 minutes with a standard deviation of 3 minutes. Assume that the data is normally distributed. What is the probability that the trip will take more than half an hour

Answer 1

Answer on Question #71163 – Math – Statistics and Probability

Question

A student commutes daily from his house to school. On average, the trip one way takes 24 minutes with a standard deviation of 3 minutes. Assume that the data is normally distributed. What is the probability that the trip will take more than half an hour.

Solution

In this dataset,

$$\mu = 24 \text{ and } \sigma = 3.$$

We need to find the probability that the trip will take more than 30 minutes.

This is

$$P(T > 30) = P(T - 24 > 30 - 24) = P(T - 24 > 6) = P((T - 24)/3 > 6/3) = P((T - 24)/3 > 2).$$

Now, if T follows N(24,9), then (T-24)/3 follows N(0,1), or the standard normal distribution.

$$P(T > 30) = 1 - (P((T - 24)/3 < 2) = 1 - \Phi(2),$$

where $\Phi$ is the cumulative distribution function of the standard normal distribution.

From the standard normal distribution tables, we have

$$\Phi(2) = 0.97725.$$

Therefore,

$$P(T > 30) = 1 - 0.97725 = 0.02275.$$

Answer: 0.02275.

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