Question

Waiting times for an order at Starbucks for all drive-through customers in the US have a uniform distribution from 3 min to 11 min (mean = 7 min, standard deviation = 2.3 min). What distribution would you use to find the probability that a randomly selected Starbucks drive-through customer in the US waits at most 9 minutes to receive their order?

Answer 1

Answer:

$P(X <9)$

And we can use the cumulative distribution function given by:

$F(X) = \frac{x-a}{b-a}, a\leq X \leq b$

And using this we got:

$P(X <9) = F(9) = \frac{9-3}{11-3}= 0.75$

Step-by-step explanation:

For this case we assume that X= represent the waiting times and for this case we have the following distribution:

$X \sim Unif (a= 3, b =11)$

And the expected value is given by:

$\mu= E(X) = \frac{a+b}{2}= \frac{3+11}{2}=7$

And the variance is given by:

$Var(X) \sigma^2 = \frac{(b-a)^2}{12} = \frac{(11-3)^2}{12} = 5.333$

And we can find the deviation like this:

$Sd(X) = \sqrt{5.333}= 2.309$

And we want to find this probability:

$P(X <9)$

And we can use the cumulative distribution function given by:

$F(X) = \frac{x-a}{b-a}, a\leq X \leq b$

And using this we got:

$P(X <9) = F(9) = \frac{9-3}{11-3}= 0.75$

Answer 2

The probability that a randomly selected Starbucks drive-through customer in the US waits at most 9 minutes to receive their order is 0.75.

What is uniform distribution?

Uniform distributions are probability distributions in which all events are equally likely to occur.

Let's assume that X represents the waiting time.

As the distribution is the uniform distribution, we can write a =3 and b =11

Now, the expected value can be written as,

$\mu = E(X) = \dfrac{a+b}{2} = \dfrac{3+11}{2}= 7$

the variance of the distribution can be written as,

$Var(X) = \sigma^2 = \dfrac{(b-a)^2}{12} = \dfrac{(11-3)^2}{12} =5.34$

And, the standard deviation can be written as,

$\sigma = \sqrt{5.34}=2.309$

In order to calculate the probability, we will use the cumulative distribution function. The cumulative function is given by the formula,

$F(X) = \dfrac{x-a}{b-a}, a\leq X\leq b$

Using the function we can get the probability as,

$F(X < 9) = F(9) = \dfrac{9-3}{11-3} = 0.75$

Hence,  the probability that a randomly selected Starbucks drive-through customer in the US waits at most 9 minutes to receive their order is 0.75.

Learn more about Uniform Distribution:

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