Question

Since the area under the normal curve within two standard deviations of the mean is 0.95, the area under the normal curve that corresponds to values greater than 2 standard deviations above the mean is 0.05.
a. True
b. False

Answer 1

Answer:

$X \sim N (\mu ,\sigma)$

And for this case we know this condition:

$P(\mu-2\sigma$

By the complement rule we know that:

$P(X< \mu -2\sigma \cup X>\mu +2\sigma) = 1-0.95=0.05$

But since the distribution is symmetrical we know that:

$P(X\mu +2\sigma) = 0.025$

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:

$X \sim N (\mu ,\sigma)$

And for this case we know this condition:

$P(\mu-2\sigma$

By the complement rule we know that:

$P(X< \mu -2\sigma \cup X>\mu +2\sigma) = 1-0.95=0.05$

But since the distribution is symmetrical we know that:

$P(X\mu +2\sigma) = 0.025$

So then the statement for this case is FALSE.

b. False