Respuesta :
Answer:
z(195) = (195-200)/50 = -0.1
z(205) = (205-200)/50 = +0.1
P(195 < x < 205) = p(-0.1 < z < 0.1) = 0.0797
Step-by-step explanation:
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Using the normal distribution and the central limit theorem, it is found that there is a 1 = 100% probability that the sample mean will be within 65 of the population mean.
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 200, hence [tex]\mu = 200[/tex].
- The standard deviation is of 50, hence [tex]\sigma = 50[/tex].
- The sample size is of 100, hence [tex]n = 100, s = \frac{50}{\sqrt{100}} = 5[/tex].
We want the probability that the sample mean will be within 65 of the population mean, hence:
[tex]Z = \frac{65}{s}[/tex]
[tex]Z = \frac{65}{5}[/tex]
[tex]Z = 13[/tex]
The probability is P(|Z| < 13), which is the p-value of Z = 13 subtracted by the p-value of Z = -13.
Looking at the z-table:
- Z = 13 has a p-value of 1.
- Z = -13 has a p-value of 0.
1 - 0 = 1
1 = 100% probability that the sample mean will be within 65 of the population mean.
To learn more about the normal distribution and the central limit theorem, you can take a look at https://brainly.com/question/24663213
