This assignment is about transforming statistics into Z-scores (and soon t-scores), and then calculating the probability of being above or below that score, or between two scores.

Determine the Z-score: Calculate the Z-score of the given value using the formula:

Z=X

where X is the value, is the mean, and is the standard deviation.

2. Convert Z-score to percentile: Once you have the Z-score, you can look up the corresponding percentile in a standard normal distribution table. For instance, a Z-score of 1.0 corresponds roughly to the 84th percentile, meaning approximately 84% of the data falls below that value.

3. Calculate percentages below and above: Since the normal distribution is symmetric, you can infer the percentage of data points below and above the given value based on its percentile. For example, if you're at the 84th percentile, roughly 84% of the data is below you and 16% is above you.

4. Compute percentages between two values: If you have two Z-scores, you can calculate the percentage of data points between them by finding the area under the curve between those Z-scores.

Let's illustrate this with an example:

Suppose you have a height of 170 cm, and the mean height for your age group is 165 cm with a standard deviation of 5 cm.

Calculate the Z-score:

Z=5170165=1

Using a Z-table, we find that the percentile for Z=1 is approximately 84.13%. This means that about 84.13% of your age group were below you, and 15.87% were above you.

And if you want to find the percent between you and the average, you would subtract the 50th percentile (the mean) from your percentile:

84.13%50%=34.13%

So, approximately 34.13% of your age group were between you and the average.

Now answer this question

Think of the growth charts when you went to the doctor as a child. What percentile were you in? What percent of your age group were below you or above you? What percent were between you and the average?These are the types of calculations we need to do with Z-(or t-)scores. We willtranslate anyvalue into its distance from the mean, plus or minus, and then we can say what percentile it is. 50% will be below the mean, and 50% above in our normal curves.