Question

Students in a large psychology class measured the time, in seconds, it took each of them to perform a certain task. The times were later converted to minutes. If a student had a standardized score of z equals one point seven two before the conversion, what is the standardized score for the student after the conversion?

Answer 1

Answer:

The standardized score for the student after the conversion is 1.72.

Step-by-step explanation:

If $X\sim N(\mu,\ \sigma^{2})$, then $Z=\frac{X-\mu}{\sigma}$, is a standard normal variate with mean, E (Z) = 0 and Var (Z) = 1. That is, $Z \sim N (0, 1)$.

The procedure of standardization transforms individual scores to standard scores for which we know the percentiles (if the data are normally distributed).  

Standardization does this by transforming individual scores from different normal distributions to a common normal distribution with a known mean, standard deviation, and percentiles.

The distribution of these z-variate is known as the standard normal distribution.

The z-scores do not have any units since the units are cancelled out in the computation of the value of z-scores.

It is provided that the data collected by the students are in seconds. Then the data were converted into minutes.

Even on converting the units the z-score value will remain same.

Consider an example:

If one reading of a student is, X = 220 seconds.

The mean and standard deviation are, μ = 120 seconds and σ = 50 seconds.

Compute the z-score as follows:

$Z=\frac{X-\mu}{\sigma}=\frac{220-120}{50}=2$

Now convert the values into minutes as follows:

X = 3.67 minutes

μ = 2 minutes

σ = 0.83 minutes

Compute the z-score as follows:

$Z=\frac{X-\mu}{\sigma}=\frac{3.67-2}{0.83}=2.005\approx 2$

The z-score is same for both the case.

Thus, the standardized score for the student after the conversion is 1.72.