Question

Find the z-score for a normal measurement that satisfies each of the following statements. (a) The point z with 10 percent of the observations falling below it.

Answer 1

Answer:

The point z with 10 percent of the observations falling below it is approximately z = -1.28.

Step-by-step explanation:

We can obtain the probabilities for any normal distributed data using a standard normal distribution, which has a mean = 0 and a standard deviation = 1.

To achieve that, we need to use z-scores. A z-score is a 'transformation' for any raw score coming from any normally distributed population. With this 'transformation' we can consult a standard normal table. This table usually presents the cumulative probability for values less than z. For instance, P(z< 0) = 0.50000.

The formula for z-scores is as follows:

$\\ z = \frac{x - \mu}{\sigma}$ (1)

Where

$\\ x\;is\;the\;raw\;score$.

$\\ \mu\;is\;the\;population\;mean$.

$\\ \sigma\;is\;the\;population\;standard\;deviation$.

We already know that the z-scores give us the distance, in standard deviations, for a 'transformed' raw score from the mean, i.e., a score of z = 1.0 tells us that the value is 1.0 standard deviation above the mean. Conversely, a z = -1.0 tells us that the value is one standard deviation below the mean (notice the negative sign).

Since the values for the cumulative standard normal table are for positive values of z, that is, they are scores above the mean, we can find those related with values below the mean thanks to the symmetrical property of the normal distribution.

In this way, to find the point z with 10 percent of the observations falling below it, if we consult the standard normal table, we know that the z score where approximately 90% of the cases are below it is approximately z = 1.28, since P(z<1.28) = 0.89973 or almost 0.90000 and the rest 10% is above it or P(z>1.28) has a probability of almost 0.10.

Well, for a value of z = -1.28, which is below the mean (at the same distance from the mean than z = 1.28, but in the opposite direction), we can determine by analogy that 10% of the values are below it and 90% are  above it (because of the symmetry of the normal distribution).

In fact, a z = -1.28 represents the 10th percentile (below it are the 10% of the values, and above it, the remaining 90% of the values) for the standard normal distribution as well as z = 1.28 is the 90th percentile for the same distribution (below it are the 90% of the values and above it are the rest 10% of them).

Mathematically, we can say that (taking into account the symmetry of the normal distribution again):

$\\ P(z$

$\\ P(z1.28)$

$\\ P(z1.28)$

Then

$\\ P(z$

Thus, the point z with 10 percent of the observations falling below it is approximately z = -1.28.

The graph below represents the values below z = -1.28 (about 10%) and those above z = 1.28 (about 10%). Notice the symmetry of this standard normal distribution, with mean = 0 and standard deviation = 1.