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
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:
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 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.