At Westonci.ca, we make it easy to get the answers you need from a community of informed and experienced contributors. Discover detailed answers to your questions from a wide network of experts on our comprehensive Q&A platform. Connect with a community of professionals ready to help you find accurate solutions to your questions quickly and efficiently.
Sagot :
Using the normal distribution and the central limit theorem, it is found that there is a 0.1896 = 18.96% probability that the average waiting time for a random sample of ten customers is between 4.0 and 4.2 minutes.
In a normal distribution with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the z-score of a measure X is given by:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
- It 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, which is the percentile of X.
- By the Central Limit Theorem, the sampling distribution of sample means of size n has standard deviation [tex]s = \frac{\sigma}{\sqrt{n}}[/tex].
In this problem:
- The mean is of 4.1 minutes, hence [tex]\mu = 4.1[/tex].
- The standard deviation is of 1.3 minutes, hence [tex]\sigma = 1.3[/tex].
- Sample of 10 customers, hence [tex]n = 10, s = \frac{1.3}{\sqrt{10}} = 0.411[/tex]
The probability is the p-value of Z when X = 4.2 subtracted by the p-value of Z when X = 4, hence:
X = 4.2
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
By the Central Limit Theorem
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{4.2 - 4.1}{0.411}[/tex]
[tex]Z = 0.24[/tex]
[tex]Z = 0.24[/tex] has a p-value of 0.5948.
X = 4
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{4 - 4.1}{0.411}[/tex]
[tex]Z = -0.24[/tex]
[tex]Z = -0.24[/tex] has a p-value of 0.4052.
0.5948 - 0.4052 = 0.1896
0.1896 = 18.96% probability that the average waiting time for a random sample of ten customers is between 4.0 and 4.2 minutes.
A similar problem is given at https://brainly.com/question/25779119
We hope you found this helpful. Feel free to come back anytime for more accurate answers and updated information. Thanks for stopping by. We strive to provide the best answers for all your questions. See you again soon. We're glad you visited Westonci.ca. Return anytime for updated answers from our knowledgeable team.
