a local content center reports that it has been experiencing a 15% rate of no shows on advanced reservations. Among 150 advanced reservations find the probability that there will be fewer than 20 no shows round the standard deviation to three decimal to work the problem
a local content center reports that it has been experiencing a 15% rate of no shows on advanced reservation Among 150 advanced reservations find the probability that there will be fewer than 20 no shows round the standard deviation to three decimal to work the problem
To find the probability that there will be fewer than 20 no shows, we need to use the binomial distribution. The standard deviation of a binomial distribution is given by the formula:
σ = √(n * P * (1 - P))
where
n = number of trials (advanced reservations)
P = probability of success (no show rate)
In this case, the no show rate is given as 15%, which can be represented as a decimal as 0.15. So, P = 0.15.
The number of trials (advanced reservations) is given as 150, so n = 150.
To calculate the standard deviation, we substitute the values into the formula:
σ = √(150 * 0.15 * (1 - 0.15))
= √(150 * 0.15 * 0.85)
= √19.275
Rounding this to three decimal places, the standard deviation is approximately 4.393.
Now, to find the probability of having fewer than 20 no shows, we can use a cumulative binomial distribution function. The formula for the cumulative distribution function is:
P(X ≤ k) = Σ (n choose x) * P^x * (1 − P)^(n−x)
where
X = the random variable (number of no shows)
k = the desired number of successes (fewer than 20 no shows)
Using this formula, we can calculate the probability. However, manually calculating it can be time-consuming. Instead, we can use statistical software or an online binomial calculator to find the probability.
Let's assume P(x ≤ 20) is the probability of having less than or equal to 20 no shows. You can use any binomial calculator available online and enter the values as follows:
- Number of trials (n): 150
- Probability of success (P): 0.15
- Number of successes (k): 20
The binomial calculator will give you the desired probability, which can be rounded to the given decimal places.
Please note that the calculation may depend on the specific binomial calculator or software you choose to use.