Question

1. Given 𝑥 = 60; and 𝑠 = 6, find the z-score that corresponds to each of the following

scores up to two decimal places.

a. 𝑥 = 70

b. 𝑥 = 58

2. Given 𝜇 = 72; and 𝜎 = 8, find the z-score that corresponds to each of the following

scores up to two decimal places.

a. 𝑥 = 68

b. 𝑥 = 80

3. Alex scored 90 during the first periodic exam in Mathematics and 88 during the

second periodic exam. The scores in first periodic exam have a mean 𝜇 = 83 and

a standard deviation 𝜎 =9. Scores in the second periodic exam have a mean 𝜇 =

80 and a standard deviation 𝜎 = 8. In which periodic exam was his standing better,

assuming that the scores in his periodic exams are normally distributed?

4. On a final examination in Biology, the mean was 75 and the standard deviation

was 12. Determine the standard score of a student who received a score of 60

assuming that the scores are normally distributed.

5. Given: 𝜇 = 64,𝜎 = 7. What is the raw score when 𝑧 = −0.76?

Answer 1

1.

a.

$$z=\dfrac{x-\bar{x}}{s}=\dfrac{70-60}{6}=\dfrac{5}{3}\approx1.67$$

b.

$$z=\dfrac{x-\bar{x}}{s}=\dfrac{58-60}{6}=-\dfrac{2}{3}\approx-0.67$$

2.

a.

$$z=\dfrac{x-\mu}{\sigma}=\dfrac{68-72}{8}=-0.50$$

b.

$$z=\dfrac{x-\mu}{\sigma}=\dfrac{80-72}{8}=1.00$$

3. To answer this question we need to find the z score for each exam.

$$z_1=\dfrac{x_1-\mu_1}{\sigma_1}=\dfrac{90-83}{9}=\dfrac{7}{9}\approx0.7778$$

$$z_2=\dfrac{x_2-\mu_2}{\sigma_2}=\dfrac{88-80}{8}=1$$

We see that the highest z-score is 1, which means that Alex was his standing better in the second periodic exam.

4.

$$z=\dfrac{x-\mu}{\sigma}=\dfrac{60-75}{12}=-1.25$$

5.

$$z=\dfrac{x-\mu}{\sigma}=\dfrac{x-64}{7}=-0.76$$

$$x=64-0.76(7)=58.68$$