10.02 Calculations with standard and normal distributions
If Dave scores 96 in a test that has a mean score of 128 and a standard deviation of 16, what is his z-score?
Frank finishes a fun run in 156 minutes. If the mean time taken to finish the race is 120 minutes and the standard deviation is 12 minutes, what is his z-score?
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\%.
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.
Dylan scored 90\% with a z-score of 2 in English, and 78\% with a z-score of 4 in Mathematics.
In which subject was his performance better, relative to the rest of his class?
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?
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.
Find Ray's z-score.
Find Gwen's z-score.
Which of the two had a better performance relative to the other students in their classes?
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?
Find the area under the curve, to four decimal places, for each part of the standardised normal curves described below:
Calculate the percentage of standardised data, to two decimal places, that is:
Calculate the probability, to four decimal places, that a z-score is:
Using your calculator, find the value of k to four decimal places for the following probabilities:
The probability of a z-score being greater than - k and at most k is equal to 0.6123 in the standard normal distribution.
The probability of a z-score being greater than - 2.11 and at most k is equal to 0.8273 in the standard normal distribution.
If X \sim N \left(20, 5^2 \right), use your calculator to find the value of k in the following parts:
P \left(X \lt k \right) = 0.65
P \left(X \gt k \right) = 0.45
P \left(k \lt X \lt 27 \right) = 0.89
P \left(21 \lt X \lt k \right) = 0.4
Languages and Mathematics are very different disciplines, and so to compare results in the two subjects, the standard deviation is used. The mean and standard deviation of exam results in each subject are given in the table:
| Mean | Std. Deviation | |
|---|---|---|
| Languages | 60 | 7 |
| Mathematics | 67 | 8 |
A student receives a mark of 81 in Languages. How many standard deviations away from the mean is this mark?
What mark in Mathematics would be equivalent to a mark of 81 in Languages?
A student receives a mark of 86.2 in Mathematics. How many standard deviations away from the mean is this mark? Round your answer to one decimal place.
What mark in Languages would be equivalent to a mark of 86.2 in Mathematics? Round your answer to one decimal place.
For the standard normal variable X \sim N \left(0, 1\right), use a CAS to determine the following values to three decimal places:
The 0.7 quantile
The 65th percentile
The lowest score in the top 20 percent
Consider the graph of a standard normal distribution showing the 68-95-99.7 rule:
Which value is the closest to the 0.5 quantile?
Which value is the closest to the 0.84 quantile?
Which value is the closest to the 16th percentile?
Consider a normal distribution defined by X \sim N \left(50, 25\right). Use the 68-95-99.7 rule to answer the following questions:
Which value is equivalent to the 0.16 quantile?
Which value is equivalent to the 0.025 quantile?
Which value is equivalent to the 97.5th percentile?
The heights of a certain species of fully grown plants are thought to be normally distributed with a mean of 40 cm and a standard deviation of 1 cm. Use the 68-95-99.7 rule to answer the following questions:
What is the height of the shortest plant in the 84th percentile?
What is the height of the shortest plant in the 0.0015 quantile?
For a normal variable defined by X \sim N \left(100, 100\right), use a CAS to determine the following values to three decimal places:
The 0.2 quantile
The 90th percentile
The lowest score that is greater than the bottom 30 percent
If X \sim N \left(30, 4^2 \right), calculate:
The 0.5 quantile
The 0.83 quantile
The 35th percentile
A random variable is normally distributed such that X \sim N \left(50, 25\right).
Calculate the standard score if X = 58.
Using a CAS calculator or otherwise, calculate the z-score for the 0.35 quantile.
Hence, find the X value for the 0.35 quantile.
Mensa is an organisation that only accepts members who score in the 98th percentile or above in an IQ test. Explain what a person has to do to get into Mensa.
If Maximilian scores 57.6, with a z score of 2, in a test that has a standard deviation of 5.8, what was the mean score?
If Han scores 43.2, with a z score of -3, in a test that has a standard deviation of 5.6, what was the mean score?
If Luke scores 68, for a z score of - 3, in a test that has a mean score of 93.5, what was the standard deviation of the test scores?
If Saoirse scores 32.5, for a z score of - 4, in a test that has a mean score of 58.5, what was the standard deviation of the test scores?
If X \sim N \left( \mu, 100 \right), use your calculator to find \mu if P \left( \mu \leq X \leq 20 \right) = 0.3013. Round your answer to two decimal places.
If X \sim N \left( \mu, 100 \right), use your calculator to find \mu if P \left( \mu \leq X \leq 30 \right) = 0.419. Round your answer to two decimal places.
If X \sim N \left( \mu, \sigma^2 \right), use your calculator to find \mu and \sigma if P \left(X \lt 70 \right) = 0.1817 and P \left (X \lt 80 \right) = 0.9655. Round your answers to two decimal places.
If X \sim N \left( \mu, \sigma^2 \right), use your calculator to find \mu and \sigma if P \left(X \lt 12 \right) = 0.2859 and P \left (X \lt 18 \right) = 0.8677. Round your answer to two decimal places.