Consider the mean \mu, standard deviation \sigma and a value x taken from a normally distributed data set. For each of the following calculate the z-score that corresponds to x:
\mu=3, \sigma=5,x=8
\mu=9, \sigma=2, x=13
\mu=3, \sigma=4, x=15
\mu=7, \sigma=2, x=3
\mu=-2, \sigma=3, x=-5
\mu=-3, \sigma=9, x=25
\mu=-5, \sigma=2, x=-9
\mu=5.8, \sigma=8.9, x=19.15
A data set is approximately normally distributed with a mean of \overline{x} and a standard deviation of s. Calculate the z-score that corresponds to each of the following values from the data set:
\overline{x} - s
\overline{x} + 2 s
\overline{x} - 3 s
A data set is approximately normally distributed. Given the following standard deviation \sigma, x-value and z-score, calculate the mean:
\sigma=3,x=7,z-\text{score}=1
\sigma=8,x=-15,z-\text{score}=-3
\sigma=4.3,x=21.46,z-\text{score}=3.2
A data set is approximately normally distributed with a mean of 8.8. If a value of 33.12 from the data set has a z-score of 3.2, calculate the standard deviation.
A data set is approximately normally distributed with a mean of 979 and a standard deviation of 25.
Find the mean of the set of z-scores.
Find the standard deviation of the set of z-scores.
A data set is approximately normally distributed with a known mean and standard deviation.
Find the mean of the set of z-scores.
Find the standard deviation of the set of z-scores.
Consider the following z-scores from a data set, which is approximately normally distributed:
State the number of standard deviations the value is away from the mean.
State if the value is above or below the mean.
z-score =3
z-score =- 2
z-score =4.29
Consider the set of marks given below:
52, \quad 56,\quad 63, \quad 66,\quad 70,\quad 73,\quad 88,\quad 88,\quad 94, \quad 95
Calculate the mean.
Calculate the population standard deviation.
If a student scored 85, find his z-score.
For each of the following examples, find the z-score:
Dave scores 96 in a test. The mean score is 128 and the standard deviation is 16.
Frank finishes a fun run in 156 minutes. The mean time is 120 minutes and the standard deviation is 12 minutes.
Christa is 157 \text{ cm} tall. The mean height in her class is 141 \text{ cm} and the standard deviation is 8 \text{ cm}.
A particular investment fund has returned 17.2\% p.a. over a period. The mean return was 8\% p.a. and the standard deviation was 2.3\%.
Maximilian scores 57.6, with a z score of 2 in a test that has a standard deviation of 5.8. Calculate the mean score in the test.
The amount of food (in kilograms) that goes to waste at a particular restaurant each week is approximately normally distributed with a standard deviation of 2\text{ kg}. One week, the restaurant wastes 70\text{ kg} of food, which has a z-score of 4. Find the average amount of waste that the restaurant produces each week.
The results of a test are approximately normally distributed with a mean of 93.5. If Luke's test score of 68 has a z-score of - 3, find the standard deviation of the test scores.
The heights of emus are normally distributed with a mean of 194\text{ cm}. If an emu's height of 171.75\text{ cm} has a z-score of - 2.5, find the value of the standard deviation.
The number of runs scored by Tobias in each of his innings is listed below:
33, 31, 32, 30, 32, 30, 32, 30, 31, 34For each of the following, round your answers to two decimal places.
Find his batting average.
Find his population standard deviation.
Find the z-score of his final innings.
Find the z-score of his highest scoring innings.
The heights of a group of Year 12 students (in centimetres) are approximately normally distributed:
167,\quad 161,\quad 159,\quad 164,\quad 161,\quad 164,\quad 162,\quad 162,\quad 166,\quad 162
If the mean is 162.8\text{ cm }and the standard deviation is 2.32\text{ cm}, complete the following table by calculating the z-scores to two decimal places:
| \text{Heights (cm)} | 159 | 161 | 161 | 162 | 162 | 162 | 164 | 164 |
|---|---|---|---|---|---|---|---|---|
| z-\text{scores} |
Find the mean of the z-scores to the nearest integer.
Find the standard deviation of the z-scores to the nearest integer.
What can be said about the mean and standard deviation of the z-scores of normally distributed data?
The number of runs scored by Mario in each cricket match is approximately normally distributed. His runs are shown below:
45,\quad 47,\quad 45,\quad 53,\quad 41,\quad 45,\quad 54,\quad 41,\quad 47,\quad 37
Find his average number of runs to two decimal places.
Find his standard deviation to two decimal places.
Complete the table by calculating the z-scores to two decimal places:
| \text{Runs} | 37 | 41 | 41 | 45 | 45 | 45 | 47 | 47 | 53 |
|---|---|---|---|---|---|---|---|---|---|
| z-\text{scores} |
Calculate the mean of the z-scores to the nearest integer.
Calculate the standard deviation of the z-scores to the nearest integer.
A general ability test has a mean score of 100 and a standard deviation of 15.
Paul received a score of 102 in the test, find his z-score correct to two decimal places.
Georgia had a z-score of 3.13, find her score in the test, correct to the nearest integer.
The following table shows the marks obtained by a student in two subjects:
Find the mark in Science that is 2 standard deviations below the mean.
Find the mark in English that is 1.5 standard deviations above the mean
Find the mark in Science that is 0.5 standard deviations above the mean
Find the mark in English that is 1 standard deviation below the mean.
| Subject | Mean | Standard Deviation |
|---|---|---|
| \text{Science} | 89 | 11 |
| \text{English} | 72 | 12 |
The number of successful free throws a basketball team makes is approximately normally distributed with a mean of 26 and a standard deviation of 5.
Find the number of successful free throws which is 2 standard deviations above the mean.
The volume of water in a bottle is approximately normally distributed with a mean of 500\text{ mL} and a standard deviation of 0.2\text{ mL}.
Find the volume of water which is 3.5 standard deviations below the mean.
Hence, calculate the z-score that corresponds to this volume of water.
Jenny scored 81\% with a z score of - 2 in English, and 72\% with a z score of - 4 in Mathematics. In which subject was her performance better, relative to the rest of the class? Explain your answer.
Ray scored 12.49 in his test, in which the mean was 7.9 and the standard deviation was 1.7. Gwen scored 30.56 in her test, in which the mean was 20.2 and the standard deviation was 2.8.
Calculate Ray's z-score.
Calculate Gwen's z-score.
Which of the two had a better performance relative to the other test takers?
Marge scored 43 in her Mathematics exam, in which the mean score was 49 and the standard deviation was 5. She also scored 92.2 in her Philosophy exam, in which the mean score was 98 and the standard deviation was 2.
Find Marge’s z-score in Mathematics.
Find Marge’s z-score in Philosophy.
Which exam did Marge do better in relative to the rest of her class?
Kathleen scored 83.4 in her Biology exam, in which the mean score was 81 and standard deviation was and 2. She also scored 60 in her Geography exam, in which the mean score was 46 and the standard deviation was 4.
Find Kathleen’s z-score in Biology.
Find Kathleen’s z-score in Geography.
Which exam did Kathleen do better relative to the rest of her class?
Ivan scored 55.25 in his test, in which the mean score was 64.5 and the standard deviation was 2.5. Maria scored 50.22 in her test, in which the mean score was 57.9 and the standard deviation was 2.4.
Find Ivan's z-score.
Find Maria's z-score.
Whose performance was better relative to the rest of their class?
Buzz’s best time in the half marathon is 90.2 minutes, while his best time in the full marathon is 185.6 minutes. Among world class runners:
The mean time to complete the half marathon is 110 minutes with a standard deviation of 6 minutes
The mean time to complete the full marathon is 200 minutes with a standard deviation of 3 minutes
Calculate the z-score of Buzz’s best time in the half marathon.
Calculate the z-score of Buzz’s best time in the full marathon.
Did Buzz perform better in the half-marathon or full-marathon, relative to the rest of the runners?
Luke’s best time in the half marathon is 141.2 minutes, while his best time in the full marathon is 242 minutes. The mean time to complete the half marathon is 125 minutes with a standard deviation of 3 minutes among world class runners, and the mean time to complete the full marathon is 226 minutes with a standard deviation of 4 minutes among world class runners.
Calculate the z score of Luke’s best time in the half marathon.
Calculate the z-score of Luke’s best time in the full marathon.
In which event does Luke have a better 'best time', relative to the world class runners?
The table shows the batting average of all first class players at each of the cricket grounds:
| Ground | Mean score | Standard deviation | Han's average | z-score |
|---|---|---|---|---|
| \text{Gabba} | 193 | 10 | 238 | |
| \text{MCG} | 191 | 13 | 153.3 | |
| \text{WACA} | 181 | 12 | 207.4 | |
| \text{SCG} | 190 | 11 | 142.7 |
Complete the table by calculating Han's z-scores.
At which ground does he play best relative to the other first class players?
The following table shows Christa’s results in the HSC, mean mark and standard deviation in each of the subjects:
| \text{Subject} | \text{Mean score} | \text{Standard deviation} | \text{Christa's mark} | z\text{-score} |
|---|---|---|---|---|
| \text{Chemistry} | 41 | 9 | 2.3 | |
| \text{History} | 47 | 7 | 77.8 | |
| \text{Physics} | 56 | 8 | 82.4 | |
| \text{Mathematics} | 48 | 3 | 38.7 | |
| \text{English} | 60 | 2 | 58.4 |
Complete the table above by calculating the z-scores for each of Christa's marks.
State the subject in which Christa performed strongest relative to her cohort. Explain your answer.
Amelia scored 60 for her History assignment and 65 for her Philosophy assignment. The following list shows the marks of all her classmates for each of the assignments:
-
History: 29,\, 60,\, 33,\, 57,\, 38,\, 56,\, 46,\, 42,\, 48,\, 12,\, 43,\, 20,\, 45,\, 27,\, 32
-
Philosophy: 55,\, 65,\, 58,\, 60,\, 44,\, 41,\, 46,\, 85,\, 77,\, 60,\, 58,\, 76,\, 54,\, 59,\, 52
Calculate the following correct to two decimal places:
Amelia’s z-score for the History assignment
Amelia’s z-score for the Philosophy assignment
In which assignment did she perform the best?
Athletes take a lower heart rate as a sign of better fitness. Victoria and John record their heart rates in beats per minute (\text{bpm}) each time they finish a triathlon. Both their heart rates are normally distributed, with Victoria’s \mu = 170 beats, \sigma = 9 beats and John's \mu = 161 beats, \sigma = 8 beats. At the end of their most recent race, Victoria’s heart rate was 192\text{ bpm} and John’s was 172 \text{ bpm}.
Calculate Victoria’s z-score in the most recent race to one decimal place.
Calculate John’s z-score in the most recent race to one decimal place.
Hence, who’s heart rate was better in the most recent race?
A factory packages boxes of two types of cereal:
A box of Rainbow Crispies has a mean mass of 600 \text{ g} with a standard deviation of 2.2 \text{ g}
A box of Honey Combs has a mean mass of 650 \text{ g} with a standard deviation of 1.4 \text{ g}
A box of Rainbow Crispies was selected at random for quality control. It had a mass of 613.2 \text{ g}. Calculate the z-score of this box.
A box of Honey Combs was selected at random for quality control. It had a mass of 653.08 \text{ g}. Calculate the z-score of this box.
Based on the z-scores, which box of cereal is closer to the mean mass of other boxes of the same type?
The table records the lengths of fish a fisherman caught in Sydney Harbour on a particular day:
Calculate the mean length of the Yellowfish Bream correct to two decimal places.
| \text{Type} | \text{Length (cm)} |
|---|---|
| \text{Yellowfin Bream} | 28,26,28,26,29 |
| \text{Flathead} | 34,36,33,35,32,36 |
Calculate the sample standard deviation of the length of a Yellowfin bream correct to two decimal places.
The next Yellowfin bream he catches is 30.40 \text{ cm} , calculate the z-score of this fish correct to one decimal place.
Calculate the mean length of the Flathead correct to two decimal places.
Calculate the standard deviation of the length of a Flathead correct to two decimal places.
The next Flathead he catches is 37.33\text{ cm}, calculate the z-score of this fish correct to one decimal place.
Which of these two fish is longer, relative to their cohort?