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.
Answers
GPT-4o mini
To find the probability that the kayak-rental company will have enough kayaks on any given day, we can use the normal distribution since the number of kayaks rented out follows a normal distribution with a known mean and standard deviation.
1. **Identify the parameters:**
- Mean (\(\mu\)) = 42 kayaks
- Standard deviation (\(\sigma\)) = 4 kayaks
- Number of kayaks available = 45
2. **Convert the number of kayaks rented to a z-score:**
The z-score can be calculated using the formula:
\[
z = \frac{X - \mu}{\sigma}
\]
where \(X\) is the number of kayaks the company has (45).
\[
z = \frac{45 - 42}{4} = \frac{3}{4} = 0.75
\]
3. **Look up the z-score in a standard normal distribution table:**
A z-score of 0.75 corresponds to the probability of renting 45 kayaks or fewer. Using a z-table or calculator, we find the cumulative probability for \(z = 0.75\).
The cumulative probability for \(z = 0.75\) is approximately \(0.7734\).
4. **Convert to percentage:**
The probability that the company will have enough kayaks is 0.7734, or 77.34%.
5. **Round the answer:**
Rounding this to the nearest tenth gives us **77.3%**.
Thus, the probability that the company will have enough kayaks on any given day is approximately **77.3%**.
1. **Identify the parameters:**
- Mean (\(\mu\)) = 42 kayaks
- Standard deviation (\(\sigma\)) = 4 kayaks
- Number of kayaks available = 45
2. **Convert the number of kayaks rented to a z-score:**
The z-score can be calculated using the formula:
\[
z = \frac{X - \mu}{\sigma}
\]
where \(X\) is the number of kayaks the company has (45).
\[
z = \frac{45 - 42}{4} = \frac{3}{4} = 0.75
\]
3. **Look up the z-score in a standard normal distribution table:**
A z-score of 0.75 corresponds to the probability of renting 45 kayaks or fewer. Using a z-table or calculator, we find the cumulative probability for \(z = 0.75\).
The cumulative probability for \(z = 0.75\) is approximately \(0.7734\).
4. **Convert to percentage:**
The probability that the company will have enough kayaks is 0.7734, or 77.34%.
5. **Round the answer:**
Rounding this to the nearest tenth gives us **77.3%**.
Thus, the probability that the company will have enough kayaks on any given day is approximately **77.3%**.