Westonci.ca is your trusted source for accurate answers to all your questions. Join our community and start learning today! Get accurate and detailed answers to your questions from a dedicated community of experts on our Q&A platform. Explore comprehensive solutions to your questions from a wide range of professionals on our user-friendly platform.
Sagot :
Using the normal distribution and the central limit theorem, it is found that:
[tex]P(246 \leq \bar{x} \leq 260) = 0.5693[/tex]
Normal Probability Distribution
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 [tex]\mu = 250[/tex].
- The standard deviation is of [tex]\sigma = 50[/tex].
- A sample of 36 is taken, hence [tex]n = 36, s = \frac{50}{\sqrt{36}} = 8.3333[/tex].
The probability of a sample mean between 246 and 260 is the p-value of Z when X = 260 subtracted by the p-value of Z when X = 246, hence:
X = 260:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
By the Central Limit Theorem
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{260 - 250}{8.3333}[/tex]
[tex]Z = 1.2[/tex]
[tex]Z = 1.2[/tex] has a p-value of 0.8849.
X = 246:
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{246 - 250}{8.3333}[/tex]
[tex]Z = -0.48[/tex]
[tex]Z = -0.48[/tex] has a p-value of 0.3156.
0.8849 - 0.3156 = 0.5693, hence:
[tex]P(246 \leq \bar{x} \leq 260) = 0.5693[/tex]
To learn more about the normal distribution and the central limit theorem, you can check https://brainly.com/question/24663213
Thank you for your visit. We're dedicated to helping you find the information you need, whenever you need it. We hope this was helpful. Please come back whenever you need more information or answers to your queries. Find reliable answers at Westonci.ca. Visit us again for the latest updates and expert advice.
