Question

A kayak-rental company needs to replace many of its kayaks, and it wants to ensure it has enough kayaks to meet the demand from customers during the summer season. On average, it rents out 42 kayaks each day with a standard deviation of 4. The company has 45 kayaks. Use a calculator or spreadsheet program to find the probability that the company will have enough kayaks on any given day. Round the answer to the nearest tenth. (1 point) The probability that there are enough kayaks on any given day is □.

Answer 1

To find the probability that the kayak-rental company will have enough kayaks on any given day, we can model the daily demand for kayaks as a normally distributed random variable. The company rents out, on average, 42 kayaks per day, with a standard deviation of 4.

We need to calculate the probability that the number of kayaks rented out (random variable \(X\)) is less than or equal to 45 (the number of kayaks available).

1. **Identify the parameters:**
- Mean (\(\mu\)) = 42
- Standard deviation (\(\sigma\)) = 4
- Number of kayaks available = 45

2. **Standardize the variable using the z-score formula:**
\[
Z = \frac{X - \mu}{\sigma}
\]
For \(X = 45\):
\[
Z = \frac{45 - 42}{4} = \frac{3}{4} = 0.75
\]

3. **Use a standard normal distribution table or calculator to find the probability associated with \(Z = 0.75\).**
Looking up \(Z = 0.75\):

The cumulative probability for \(Z = 0.75\) is approximately \(0.7734\).

4. **Interpret the result:**
This means that the probability that the company will have enough kayaks on any given day is approximately \(0.7734\) or \(77.34\%\).

5. **Round to the nearest tenth:**
The probability that there are enough kayaks on any given day is **0.8** when rounded to the nearest tenth.

Therefore, the answer is:

The probability that there are enough kayaks on any given day is **0.8**.