Question

1. scores on a history test have an average of 80 with a standard deviation of 6. what is the z - score for a student who earned a 75 on the test?
2. the weight of chocolate bars from a particular chocolate factory has a mean of 8 ounces with a standard deviation of 0.1 ounces. what is the z - score corresponding to a weight of 8.17 ounces?
3. books in the library are found to have an average length of 350 pages with a standard deviation of 100 pages. what is the z - score corresponding to a book of length 80 pages?
4. the temperature is recorded at 60 airports in a region. the average temperature is 67 degrees fahrenheit with a standard deviation of 5 degrees. what is the z - score for a temperature of 68 degrees?
5. a group of friends compares what they received while trick or treating. they find that the average number of pieces of candy received is 43, with a standard deviation of 2. what is the z - score corresponding to 20 pieces of candy?

Explanation:

Step1: Recall z - score formula

The formula for the z - score is $z=\frac{x-\mu}{\sigma}$, where $x$ is the value, $\mu$ is the mean, and $\sigma$ is the standard deviation.

Step2: Solve problem 1

Given $\mu = 80$, $\sigma=6$, and $x = 75$. Substitute into the formula: $z=\frac{75 - 80}{6}=\frac{- 5}{6}\approx - 0.83$.

Step3: Solve problem 2

Given $\mu = 8$, $\sigma = 0.1$, and $x=8.17$. Substitute into the formula: $z=\frac{8.17 - 8}{0.1}=\frac{0.17}{0.1}=1.7$.

Step4: Solve problem 3

Given $\mu = 350$, $\sigma = 100$, and $x = 80$. Substitute into the formula: $z=\frac{80 - 350}{100}=\frac{-270}{100}=-2.7$.

Step5: Solve problem 4

Given $\mu = 67$, $\sigma = 5$, and $x = 68$. Substitute into the formula: $z=\frac{68 - 67}{5}=\frac{1}{5}=0.2$.

Step6: Solve problem 5

Given $\mu = 43$, $\sigma = 2$, and $x = 20$. Substitute into the formula: $z=\frac{20 - 43}{2}=\frac{-23}{2}=-11.5$.

Answer:

1. $z\approx - 0.83$
2. $z = 1.7$
3. $z=-2.7$
4. $z = 0.2$
5. $z=-11.5$