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 graphing calculator to find the probability that the company will have enough kayaks on any given day.

Round the answer to the nearest tenth.

(2 points)
Responses

95.4%
95.4%

84.1%
84.1%

77.3%
77.3%

68.3%
68.3%

Answer 1

To find the probability that the company will have enough kayaks on any given day, we can model the daily kayak rentals as a normal distribution since the average number of rentals and the standard deviation are given.

1. **Identify the parameters of the normal distribution:**
- Mean (μ) = 42 kayaks
- Standard deviation (σ) = 4 kayaks
- The company has 45 kayaks available.

2. **Calculate the z-score for the maximum number of rentals (45 kayaks):**

The z-score can be calculated using the formula:
\[
z = \frac{X - \mu}{\sigma}
\]
where:
- \(X\) is the number we are comparing (in this case, 45 kayaks),
- \(\mu\) is the mean (42 kayaks),
- \(\sigma\) is the standard deviation (4 kayaks).

Substituting the values into the formula:
\[
z = \frac{45 - 42}{4} = \frac{3}{4} = 0.75
\]

3. **Find the probability corresponding to the z-score using a z-table** (or a calculator):

Looking up the z-score of 0.75, we find that the area to the left of this z-score (which represents the probability of renting 45 or fewer kayaks) is approximately 0.7734.

4. **Convert the probability to a percentage:**
\[
0.7734 \times 100 \approx 77.3\%
\]

Thus, the probability that the company will have enough kayaks on any given day is **77.3%**.

Based on the provided response options, the correct answer is:

**77.3%**.