Mathematics, 02.07.2020 17:01 saif27

The following questions pertain to the properties of the STANDARD NORMAL distribution. (a) True or False: The distribution is bell-shaped and symmetric. True False (b) True or False: The mean of the distribution is 0. True False (c) True or False The probability to the left of the mean is 0. True False (d) True or False The standard deviation of the distribution is 1. True False

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Answer from: sascsl2743

a. The distribution is bell-shaped and symmetric: True.

b. The distribution is bell-shaped and symmetric: True.

c. The probability to the left of the mean is 0: False.

d. The standard deviation of the distribution is 1: True.

Step-by-step explanation:

The Standard Normal distribution is a normal distribution with mean, $\\ \mu = 0$ , and standard deviation, $\\ \sigma = 1$ .

It is important to recall that the parameters of the Normal distributions, namely, $\\ \mu$ and $\\ \sigma$ characterized them.

We can use the Standard Normal distribution to find probabilities for any normally distributed data. All we have to do is normalized them through z-scores:

$\\ z = \frac{x - \mu}{\sigma}$ [1]

Where $\\ x$ is the raw score that we want to standardize.

Therefore, taking into account all this information, we can answer the following questions about the Standard Normal distribution:

(a) True or False: The distribution is bell-shaped and symmetric

True. As the normal distribution, the standard normal distribution is also bell-shape and it is symmetrical around the mean. The standardized values or z-scores, which represent the distance from the mean in standard deviations units, are the same but when it is above the mean, the z-score is positive, and negative when it is below the mean. This result is a consequence of the symmetry of this distribution respect to the mean of the distribution.

(b) True or False: The mean of the distribution is 0.

True. Since the Standard Normal uses standardized values, if we use [1], we have:

$\\ z = \frac{x - \mu}{\sigma}$

If $\\ x = \mu$

$\\ z = \frac{\mu - \mu}{\sigma}$

$\\ z = \frac{0}{\sigma}$

$\\ z = 0$

Then, the value for the mean is where z = 0. A z-score is a linear transformation of the original data. For this reason, the transformed mean is equivalent to 0 in the standard normal distribution. We only need to find distances from this zero in standard normal deviations or z-scores to find probabilities.

False. The probability to the left of the mean is not 0. The cumulative probability from $\\ -\infty$ until the mean is 0.5000 or $\\ P(-\infty < z < 0) = 0.5$ .

(d) True or False: The standard deviation of the distribution is 1.

True. The standard normal distribution is a convenient way of calculate probabilities for any normal distribution. The standardized variable, represented by [1], permits us to use one table (the standard normal table) for all normal distributions.

In this distribution, the z-score is always divided by the standard deviation of the population. Then, the standard deviation for the standard normal distribution are times or fractions of the standard deviation of the population, since we divide the distance of a raw score from the mean of the population, $\\ x - \mu$ , by it. As a result, the standard deviation for the standard normal distribution will be times (1, 2, 3, 0.96, -1, -2, etc) the standard deviation of any normal distribution, $\\ \sigma$ .

In this case, the linear transformation of the original data for one standard deviation from the mean is z = 1. Therefore, the standard deviation for the standard normal distribution is the unit.

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