Question

Suppose you observe a data point x = 12 and it is known that this data point came from a normal distribution with mean 5 and standard deviation 2. Which of the following statements is true regarding the observation of x = 12? The observation would not be considered unusual, because we know exactly which normal distribution it comes from. The observation would be considered unusual because it is farther than three standard deviations from the mean. The observation would not be considered unusual, because it comes from a normal distribution. The observation would not be considered unusual because it is only about three standard deviations from the mean. The observation would be considered unusual because x = 12 is over twice as large as the mean of the distribution.

Answer 1

Answer:

The observation would be considered unusual because it is farther than three standard deviations from the mean.

Step-by-step explanation:

Problems of normally distributed samples are solved using the z-score formula.

In a set with mean $\mu$ and standard deviation $\sigma$, the zscore of a measure X is given by:

$Z = \frac{X - \mu}{\sigma}$

The Z-score 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. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

When Z has an absolute value higher than 2, the observation is considered unusual.

In this problem, we have that:

$\mu = 5, \sigma = 2, X = 12$

$Z = \frac{X - \mu}{\sigma}$

$Z = \frac{12 - 5}{2}$

$Z = 3.5$

So the correct answer is:

The observation would be considered unusual because it is farther than three standard deviations from the mean.