The mean of a set of scores is \mu = 51, and the standard deviation is \sigma = 16. Find the value of:
\mu - \sigma
\mu + 2 \sigma
\mu + 3 \sigma
\mu - 2 \sigma
\mu + 0.5 \sigma
\mu - \dfrac{2 \sigma}{3}
The literacy rate of a population is used to help measure the level of development of a country. The average literacy rate in a particular country is 59\%, and the standard deviation is 5\%.
Find the literacy rate that is 3 standard deviations above the mean.
Find the literacy rate that is 2 standard deviations below the mean.
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 |
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
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\%.
Packets of flour are each labelled as having a mass of 1 \text{ kg}. The mass of these packets is normally distributed with a mean of 1.06 \text{ kg} and a standard deviation of 0.03 \text{ kg}.
Complete the following table:
| 1 | 1.03 | 1.06 | 1.09 | 1.12 | |
|---|---|---|---|---|---|
| z\text{-score} |
Find the percentage of packets that will have a mass less than 1.06 \text{ kg}.
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.
For each of the following examples, find x, the test score each student received:
Iain's z-score in a test is 1, the mean mark is 62\% and standard deviation is 3\%.
Rochelle's z-score in a test is - 3, the mean mark is 80\% and standard deviation is 4\%.
Aaron's z-score in a test is 3.6, the mean mark is 75\% and standard deviation is 5\%.
Sean's z-score in a test is - 1.1, the mean mark is 72\% and standard deviation is 6\%.
Consider the set of marks given below:
52, 56, 63, 66, 70, 73, 88, 88, 94, 95For each of the following, round your answers to two decimal places.
Calculate the mean.
Find the population standard deviation.
Given Dave scored 85, find his z-score.
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.
In an entrance exam, applicants completed two papers. On average, students performed better in Paper 1, but their marks were less spread out in Paper 2.
Describe how the size of the mean and standard deviation of Paper 1 would compare to Paper 2.
The following table shows the marks obtained by a student in two subjects:
How many standard deviations above the mean was the student's score in Science?
In which subject was his performance better relative to the rest of their class?
| Mark | Mean | Std. Deviation | |
|---|---|---|---|
| Science | 100 | 44 | 14 |
| Mathematics | 98 | 68 | 15 |
The mean and standard deviation of exam results in subjects French and Mathematics are given in the table:
| Mean | Std. Deviation | |
|---|---|---|
| French | 60 | 7 |
| Mathematics | 67 | 8 |
A student receives a mark of 81 in French. How many standard deviations away from the mean is this mark?
What mark in Mathematics would be equivalent to a mark of 81 in French?
A student receives a mark of 86.2 in Mathematics. How many standard deviations away from the mean is this mark?
What mark in French would be equivalent to a mark of 86.2 in Mathematics?
For each of the following scenarios, in which subject was the student's performance better, relative to the rest of their class?
Dylan scored 90\% with a z-score of 2 in English, and 78\% with a z-score of 4 in Mathematics.
Jenny scored 81\% with a z-score of - 2 in English, and 72\% with a z-score of - 4 in Mathematics.
Ray scored 12.49 in his test, in which the mean score was 7.9 and the standard deviation was 1.7. Gwen scored 30.56 in her test, in which the mean score was 20.2 and the standard deviation was 2.8.
Find Ray's z-score.
Find Gwen's z-score.
Whose performance was better relative to the rest of their class?
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?
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.
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 results of a test are approximately normally distributed with a standard deviation of 5.8. If Maximilian's test score of 57.6 has a z-score of 2, find the mean of the test scores.
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 percentage of people in each country with internet access is averaged and found to be 30\%. In one particular country, the percentage is 67.5\%, which is 2.5 standard deviations above the mean. Find the standard deviation of the distribution.
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.