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
The probability that there are enough kayaks on any given day is
Answers
GPT-4o mini
To find the probability that the kayak-rental company will have enough kayaks on any given day, we can model the daily kayak rentals as a normal distribution since it is stated that the average rented is 42 with a standard deviation of 4.
We'll denote:
- Mean (μ) = 42
- Standard Deviation (σ) = 4
- Number of kayaks available = 45
We need to find the probability that the rental demand (X) is less than or equal to 45:
\[
P(X \leq 45)
\]
To do this, we will first standardize the variable using the z-score formula:
\[
Z = \frac{X - \mu}{\sigma}
\]
Plugging in our values:
\[
Z = \frac{45 - 42}{4} = \frac{3}{4} = 0.75
\]
Now we need to find the cumulative probability for \(Z = 0.75\) using a standard normal distribution table or a calculator.
Using a standard normal distribution table:
The cumulative probability for \(Z = 0.75\) is approximately 0.7734.
This means that the probability that the company will have enough kayaks on any given day is approximately:
\[
P(X \leq 45) \approx 0.7734
\]
Rounding to the nearest tenth, the answer is:
**Probability that there are enough kayaks on any given day is 0.8.**
We'll denote:
- Mean (μ) = 42
- Standard Deviation (σ) = 4
- Number of kayaks available = 45
We need to find the probability that the rental demand (X) is less than or equal to 45:
\[
P(X \leq 45)
\]
To do this, we will first standardize the variable using the z-score formula:
\[
Z = \frac{X - \mu}{\sigma}
\]
Plugging in our values:
\[
Z = \frac{45 - 42}{4} = \frac{3}{4} = 0.75
\]
Now we need to find the cumulative probability for \(Z = 0.75\) using a standard normal distribution table or a calculator.
Using a standard normal distribution table:
The cumulative probability for \(Z = 0.75\) is approximately 0.7734.
This means that the probability that the company will have enough kayaks on any given day is approximately:
\[
P(X \leq 45) \approx 0.7734
\]
Rounding to the nearest tenth, the answer is:
**Probability that there are enough kayaks on any given day is 0.8.**