Question

Suppose several high schools in a city are sponsoring a walk-athon to raise money for a local charity. A certain route is mapped out through the city and each participant finds sponsors to donate money for the walk. If a participant completes the walk under a certain time, extra money will be donated. You are responsible for recording the finishing times for each of the participants. After collecting all of the data, you determine that the finishing times are normally distributed with a mean of 2.6 hours and a standard deviation of 0.3 hour.

Answer the following questions in complete sentences.

1. What percent of the players finished the walk in less than 2 hours? Show your work.

2. What is the probability that two randomly chosen players completed the walk in 2.9 hours or more? Show your work.

3. What is the probability that four randomly chosen players completed the walk between 1.7 and 2.9 hours? Show your work.

4. What observations can you make about the number of participants who complete the walk in more than 3.5 hours, given that there are 1,200 participants?

Answer 1

Approximately 2.28% of the players finished the walk in less than 2 hours. The probability that two randomly chosen players completed the walk in 2.9 hours or more is approximately 15.87%. The probability that four randomly chosen players completed the walk between 1.7 and 2.9 hours is approximately 15.74%.

Step-by-step explanation:

1. To find the percent of players who finished the walk in less than 2 hours, we need to calculate the z-score and use the standard normal distribution table.

First, calculate the z-score:

z = (x - μ) / σ

z = (2 - 2.6) / 0.3 = -2

Using the standard normal distribution table, we find that the area to the left of z = -2 is approximately 0.0228 or 2.28%. Therefore, approximately 2.28% of the players finished the walk in less than 2 hours.

2. To find the probability that two randomly chosen players completed the walk in 2.9 hours or more, we need to calculate the z-score for 2.9 and use the standard normal distribution table.

First, calculate the z-score:

z = (x - μ) / σ

z = (2.9 - 2.6) / 0.3 = 1

Using the standard normal distribution table, we find that the area to the right of z = 1 is approximately 0.1587 or 15.87%. Therefore, the probability that two randomly chosen players completed the walk in 2.9 hours or more is approximately 15.87%.

3. To find the probability that four randomly chosen players completed the walk between 1.7 and 2.9 hours, we need to calculate the z-scores for 1.7 and 2.9 and use the standard normal distribution table.

First, calculate the z-scores:

z1 = (x1 - μ) / σ = (1.7 - 2.6) / 0.3 = -3

z2 = (x2 - μ) / σ = (2.9 - 2.6) / 0.3 = 1

Using the standard normal distribution table, we find that the area to the left of z1 = -3 is approximately 0.0013 or 0.13% and the area to the right of z2 = 1 is approximately 0.1587 or 15.87%. Therefore, the probability that four randomly chosen players completed the walk between 1.7 and 2.9 hours is approximately (0.13% - 15.87%) = 15.74%.

4. Given that there are 1,200 participants, we can make the observation that the number of participants who complete the walk in more than 3.5 hours is likely to be very small. This is because the mean is 2.6 hours and the standard deviation is 0.3 hour, indicating that the majority of participants finish the walk within a narrow time range around the mean.