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 a given day, we can model the daily demand for kayaks using a normal distribution.
Given:
- The average number of kayaks rented per day (mean, μ) = 42
- The standard deviation of kayaks rented (σ) = 4
- The number of kayaks available = 45
We need to find the probability that the number of kayaks rented is less than or equal to the number of kayaks available (45). This means we need to calculate \(P(X \leq 45)\), where \(X\) is a normally distributed random variable with mean μ and standard deviation σ.
1. **Standardize the variable**:
To find this probability, we first need to convert 45 to a z-score using the formula:
\[
Z = \frac{X - \mu}{\sigma}
\]
where \(X = 45\), \(μ = 42\), and \(σ = 4\).
\[
Z = \frac{45 - 42}{4} = \frac{3}{4} = 0.75
\]
2. **Find the probability**:
Next, we will use the z-score to find the corresponding probability from the standard normal distribution. Using a z-table or a calculator for the normal distribution, we can find \(P(Z \leq 0.75)\).
The corresponding probability for \(Z = 0.75\) is approximately 0.7734.
3. **Round the answer**:
Finally, round the answer to the nearest tenth:
\[
P(X \leq 45) \approx 0.7734 \quad \text{(or 77.3% when expressed as a percentage)}
\]
Thus, the probability that the company will have enough kayaks on any given day is approximately **0.8** when rounded to the nearest tenth.
Therefore, the final answer is:
**The probability that there are enough kayaks on any given day is 0.8.**
Given:
- The average number of kayaks rented per day (mean, μ) = 42
- The standard deviation of kayaks rented (σ) = 4
- The number of kayaks available = 45
We need to find the probability that the number of kayaks rented is less than or equal to the number of kayaks available (45). This means we need to calculate \(P(X \leq 45)\), where \(X\) is a normally distributed random variable with mean μ and standard deviation σ.
1. **Standardize the variable**:
To find this probability, we first need to convert 45 to a z-score using the formula:
\[
Z = \frac{X - \mu}{\sigma}
\]
where \(X = 45\), \(μ = 42\), and \(σ = 4\).
\[
Z = \frac{45 - 42}{4} = \frac{3}{4} = 0.75
\]
2. **Find the probability**:
Next, we will use the z-score to find the corresponding probability from the standard normal distribution. Using a z-table or a calculator for the normal distribution, we can find \(P(Z \leq 0.75)\).
The corresponding probability for \(Z = 0.75\) is approximately 0.7734.
3. **Round the answer**:
Finally, round the answer to the nearest tenth:
\[
P(X \leq 45) \approx 0.7734 \quad \text{(or 77.3% when expressed as a percentage)}
\]
Thus, the probability that the company will have enough kayaks on any given day is approximately **0.8** when rounded to the nearest tenth.
Therefore, the final answer is:
**The probability that there are enough kayaks on any given day is 0.8.**