Respuesta :
c) The probability that a sample size of 36 has a sample mean of less than 19 is of: 0.0228 = 2.28%.
d) It would be unusual for a random sample of size 36 from the x distribution to have a sample mean less than 19, as the z-score has an absolute value of 2.
How to obtain probabilities using the normal distribution?
The z-score of a measure X of a variable that has mean symbolized by [tex]\mu[/tex] and standard deviation symbolized by [tex]\sigma[/tex] is obtained by the rule presented as follows:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
- The z-score represents how many standard deviations the measure X is above or below the mean of the distribution, depending if the obtained z-score is positive or negative.
- Using the z-score table, the p-value associated with the calculated z-score is found, and it represents the percentile of the measure X in the distribution.
- An unusual measure in the normal distribution has a z-score with absolute value of at least 2.
- By the Central Limit Theorem, the sampling distribution of sample means of size n has standard deviation [tex]s = \frac{\sigma}{\sqrt{n}}[/tex].
The parameters of this problem are given as follows:
[tex]\mu = 20, \sigma = 3, n = 36, s = \frac{3}{\sqrt{36}} = 0.5[/tex]
The probability that a sample size of 36 has a sample mean of less than 19 is the p-value of Z when X = 19, hence:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
By the Central Limit Theorem
[tex]Z = \frac{X - \mu}{s}[/tex]
Z = (19 - 20)/0.5
Z = -2
Z = -2 has a p-value of 0.0228.
Missing Information
Suppose x has a distribution with a mean of 20 and a standard deviation of 3. Random samples of size n=36 are drawn.
C. Find P(x<19) (answer should be: 0.0228)
D. Would it be unusual for a random sample of size 36 from the x distribution to have a sample mean less than 19? Explain. (answer should be: Yes, only about 2.3% of all such samples have means less than 19)
More can be learned about the normal distribution at https://brainly.com/question/25800303
#SPJ1
