Respuesta :
Answer:
[tex]X \sim N (\mu ,\sigma)[/tex]
And for this case we know this condition:
[tex]P(\mu-2\sigma <X < \mu +2\sigma) =0.95[/tex]
By the complement rule we know that:
[tex] P(X< \mu -2\sigma \cup X>\mu +2\sigma) = 1-0.95=0.05[/tex]
But since the distribution is symmetrical we know that:
[tex] P(X<\mu-2\sigma)= P(X>\mu +2\sigma) = 0.025[/tex]
So then the statement for this case is FALSE.
b. False
Step-by-step explanation:
Previous concepts
Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".
The Z-score is "a numerical measurement used in statistics of a value's relationship to the mean (average) of a group of values, measured in terms of standard deviations from the mean".
Solution to the problem
For this case if we define the random variable of interest X and we know that this random variable follows a normal distribution:
[tex]X \sim N (\mu ,\sigma)[/tex]
And for this case we know this condition:
[tex]P(\mu-2\sigma <X < \mu +2\sigma) =0.95[/tex]
By the complement rule we know that:
[tex] P(X< \mu -2\sigma \cup X>\mu +2\sigma) = 1-0.95=0.05[/tex]
But since the distribution is symmetrical we know that:
[tex] P(X<\mu-2\sigma)= P(X>\mu +2\sigma) = 0.025[/tex]
So then the statement for this case is FALSE.
b. False
Using the symmetry of the normal distribution, it is found that this statement is False, option b.
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.
- The normal distribution is symmetric, which means that 50% of the measures are below the mean and 50% are above.
- 95% of the measures are within 2 standard deviations of the mean, which means that this area is between the 2.5th percentile and the 97.5th percentile.
- Of the other 5%, 2.5% is between the 0th percentile and the 2.5th percentile, and 2.5% is between the 97.5th percentile and the 100th percentile.
- Thus, the area under the normal curve that corresponds to values greater than 2 standard deviations above the mean is 0.025, which is corroborated by the p-value of Z = 2 being close to 0.975, and the area above is 1 - 0.975 = 0.025. Thus, the statement is False, and option b is correct.
A similar problem is given at https://brainly.com/question/14243195
