Answer:
5) 203.35
Step-by-step explanation:
To solve this question, we need to understand the normal probability distribution and the central limit theorem.
Normal Probability Distribution:
Problems of normal distributions can be solved using the z-score formula.
In a set with mean
and standard deviation
, the z-score of a measure X is given by:
![Z = \frac{X - \mu}{\sigma}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7BX%20-%20%5Cmu%7D%7B%5Csigma%7D)
The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the p-value, we get the probability that the value of the measure is greater than X.
Central Limit Theorem
The Central Limit Theorem establishes that, for a normally distributed random variable X, with mean
and standard deviation
, the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean
and standard deviation
.
For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.
The average service time for McDonald's is 203.21 seconds with a standard deviation of 5.67 seconds.
This means that ![\mu = 203.21, \sigma = 5.67](https://tex.z-dn.net/?f=%5Cmu%20%3D%20203.21%2C%20%5Csigma%20%3D%205.67)
Sample of 90
This means that ![n = 90, s = \frac{5.67}{\sqrt{90}} = 0.59767](https://tex.z-dn.net/?f=n%20%3D%2090%2C%20s%20%3D%20%5Cfrac%7B5.67%7D%7B%5Csqrt%7B90%7D%7D%20%3D%200.59767)
There is a 51% chance that the average drive-thru service time is less than ________ seconds.
X when Z is in the 51st percentile, that is, has a p-value of 0.51, so X when Z = 0.025.
![Z = \frac{X - \mu}{\sigma}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7BX%20-%20%5Cmu%7D%7B%5Csigma%7D)
By the Central Limit Theorem
![Z = \frac{X - \mu}{s}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7BX%20-%20%5Cmu%7D%7Bs%7D)
![0.025 = \frac{X - 203.21}{0.59767}](https://tex.z-dn.net/?f=0.025%20%3D%20%5Cfrac%7BX%20-%20203.21%7D%7B0.59767%7D)
![X - 203.32 = 0.025*0.59767](https://tex.z-dn.net/?f=X%20-%20203.32%20%3D%200.025%2A0.59767)
![X = 203.35](https://tex.z-dn.net/?f=X%20%3D%20203.35)
203.35 seconds.