Z-Scores

Show your work for each problem.

1.
Test scores have mean $\mu = 50$ and standard deviation $\sigma = 5$ . What is the z-score of a score of $60$ ?
1. a)-2
2. b)10
3. c)2
4. d)0.5
Answer: 2
$z = \frac{60 - 50}{5} = \frac{10}{5} = 2$ . The score is 2 standard deviations above the mean. $10$ is only the distance in points.
2.
A distribution has $\mu = 40$ and $\sigma = 4$ . Find the z-score of $x = 46$ .
Answer: 1.5
$z = \frac{46 - 40}{4} = \frac{6}{4} = 1.5$ : the value is 1.5 standard deviations above the mean.
3.
A value $x = 18$ comes from a distribution with $\mu = 12$ and $\sigma = 3$ . Find its z-score.
Answer: 2
1. What is $x - \mu = 18 - 12$ ? 6
2. Divide by $\sigma = 3$ . What is z? 2
4.
A value has a z-score of $z = -1.5$ . What does that mean?
1. a)It is 1.5 standard deviations below the mean
2. b)It is 15% below the mean
3. c)It is 1.5 points below the mean
4. d)It is 1.5 standard deviations above the mean
Answer: It is 1.5 standard deviations below the mean
A z-score counts standard deviations, not points or percent. The minus sign means below the mean, so $z = -1.5$ is 1.5 standard deviations below the mean.
5.
Adult weights in a group have $\mu = 65$ kg and $\sigma = 10$ kg. What is the z-score of a person who weighs $47$ kg?
Answer: -1.8
$z = \frac{47 - 65}{10} = \frac{-18}{10} = -1.8$ : 1.8 standard deviations below the mean.
6.
A distribution has $\mu = 60$ and $\sigma = 5$ . Which value has the z-score $z = -2$ ?
Answer: 50
1. What is $z\sigma = (-2) \times 5$ ? -10
2. What is $x = \mu + z\sigma = 60 + (-10)$ ? 50
7.
Four students wrote different tests. Who did best compared with the others who wrote the same test?
1. a)Ben: 85 points (μ = 75, σ = 10)
2. b)Ana: 78 points (μ = 70, σ = 4)
3. c)Cleo: 90 points (μ = 84, σ = 6)
4. d)Dan: 66 points (μ = 60, σ = 5)
Answer: Ana: 78 points (μ = 70, σ = 4)
Ana: $\frac{78 - 70}{4} = 2$ . Ben: $\frac{85 - 75}{10} = 1$ . Cleo: $\frac{90 - 84}{6} = 1$ . Dan: $\frac{66 - 60}{5} = 1.2$ . Ana has the largest z-score, even though Cleo has the most points.
8.
Scores on a national test have $\mu = 500$ and $\sigma = 100$ . Which score has the z-score $z = -1.2$ ?
Answer: 380
$x = 500 + (-1.2)(100) = 500 - 120 = 380$ .
9.
Nora runs 5 km in $24$ min (her club: $\mu = 27$ min, $\sigma = 2$ min) and swims 100 m in $75$ s (her club: $\mu = 80$ s, $\sigma = 4$ s). Lower times are better. In which sport is she relatively better? Give the z-score of that sport.
Answer: -1.5
1. What is her running z-score $\frac{24 - 27}{2}$ ? -1.5
2. What is her swimming z-score $\frac{75 - 80}{4}$ ? -1.25
3. Lower is better, so the smaller z wins. Type the smaller z-score. -1.5
10.
IQ scores have $\mu = 100$ and $\sigma = 15$ . Which score is unusual ( $|z| > 2$ )?
1. a)75
2. b)85
3. c)134
4. d)120
Answer: 134
$z(120) \approx 1.33$ , $z(134) = \frac{34}{15} \approx 2.27$ , $z(85) = -1$ , $z(75) \approx -1.67$ . Only 134 has $|z| > 2$ ; the usual range is 70 to 130.
11.
In a school's 100 m race the times have $\mu = 13.5$ s and $\sigma = 0.6$ s. Find the z-score of a time of $12.6$ s.
Answer: -1.5
$z = \frac{12.6 - 13.5}{0.6} = \frac{-0.9}{0.6} = -1.5$ . In a race that is a good result: 1.5 standard deviations faster than average.
12.
A phone battery lasts on average $\mu = 40$ hours with $\sigma = 3$ hours. One battery lasts only $33.4$ hours. Is that unusual?
Answer: -2.2
1. What is $33.4 - 40$ ? -6.6
2. Divide by $\sigma = 3$ . What is z? -2.2
3. Is $|z| > 2$ , so the battery life is unusual? Yes
13.
Resting heart rates of teenagers have $\mu = 72$ beats per minute and $\sigma = 8$ . Find the range of usual values, from $z = -2$ to $z = 2$ .
Answer: [56, 88]
1. What is $2\sigma = 2 \times 8$ ? 16
2. Lower end: $72 - 16$ ? 56
3. Upper end: $72 + 16$ ? 88
14.
A teacher adds 5 bonus points to every student's score. What happens to each student's z-score?
1. a)It decreases by 5
2. b)It increases by 5/σ
3. c)It increases by 5
4. d)It stays the same
Answer: It stays the same
Every score and the mean both go up by 5, so $x - \mu$ does not change. The spread does not change either, so $\sigma$ stays the same. Therefore $z = \frac{x - \mu}{\sigma}$ stays the same.
15.
A test has mean $\mu = 80$ . A score of $92$ has the z-score $z = 1.5$ . What is the standard deviation $\sigma$ ?
Answer: 8
$1.5 = \frac{92 - 80}{\sigma} = \frac{12}{\sigma}$ , so $\sigma = \frac{12}{1.5} = 8$ .
16.
Scores on an entrance exam have $\mu = 1050$ and $\sigma = 200$ . A scholarship needs a z-score of at least $1.75$ . What is the lowest score that qualifies?
Answer: 1400
1. What is $z\sigma = 1.75 \times 200$ ? 350
2. What is $x = 1050 + 350$ ? 1400
17.
In a class, a score of $78$ has the z-score $z = 1.5$ and the standard deviation is $\sigma = 6$ . What is the class mean $\mu$ ?
Answer: 69
1. How many points above the mean is 78? Find $z\sigma = 1.5 \times 6$ . 9
2. What is $\mu = 78 - 9$ ? 69
18.
On a test, a score of $70$ has $z = -1$ and a score of $94$ has $z = 2$ . Find the standard deviation $\sigma$ and the mean $\mu$ . Give $\mu$ as the final answer.
Answer: 78
1. How many standard deviations apart are the two scores? ( $2 - (-1)$ ) 3
2. How many points apart are they? ( $94 - 70$ ) 24
3. So $3\sigma = 24$ . What is $\sigma$ ? 8
4. 70 is one standard deviation below the mean. What is $\mu = 70 + 8$ ? 78