Standardizing Measurements and the Normal Distribution Worksheet

A distribution has a mean of 50 and a standard deviation of 5. Calculate the z-score for a measurement of 60.

z = (60 - 50) / 5 = 2.0

In a distribution with mean 100 and standard deviation 15, what is the z-score of a value equal to 100?

z = (100 - 100) / 15 = 0

Test scores are normally distributed with a mean of 70 and standard deviation of 8. What is the z-score for a score of 62?

z = (62 - 70) / 8 = -1.0

Given a normal distribution with mean 80 and standard deviation 12, what proportion of values lies below 80?

0.5 (or 50%), because the normal distribution is perfectly symmetric around its mean.

A population has a mean of 170 cm and a standard deviation of 8 cm. Find the measurement that corresponds to a z-score of -2.

x = 170 + (-2 * 8) = 154 cm

A dataset has a mean of 40 and a variance of 16. What is the z-score for a value of 46?

The standard deviation is the square root of 16, which is 4. z = (46 - 40) / 4 = 1.5

If the probability of falling below z = 1.2 is 0.8849, what is the probability of falling above z = 1.2?

1 - 0.8849 = 0.1151

Using symmetry and the fact that P(Z < 1.5) = 0.9332, find the probability that a standard score is less than -1.5.

P(Z < -1.5) = P(Z > 1.5) = 1 - 0.9332 = 0.0668