A random variable, X, is normally distributed with a mean of 5 and a standard deviation of 4. Find the following probabilities to four decimal places:
P\left(X < 16 \right)
P\left(X > 16 \right)
P\left(0 < X < 16 \right)
A random variable, X, is normally distributed with a mean of - 130 and a standard deviation of 45. Find the following probabilities to four decimal places:
P \left( X < -88 \right)
P \left( X > -88 \right)
P \left(-88 < X < 34 \right)
A random variable, X, is normally distributed with a mean of 20 and a standard deviation of 9. Find the following probabilities to four decimal places:
P \left( X < 8 \right)
P \left( X > 8 \right)
P \left(8 < X < 51 \right)
A random variable, X, is normally distributed with a mean of 9 and a standard deviation of 78. Find the following probabilities to four decimal places:
P \left( X < 156 \right)
P \left( X > 156 \right)
P \left(-28 < X < 156 \right)
A random variable, X, is normally distributed with a mean of 0.8 and a standard deviation of 0.3. Find the following probabilities to four decimal places:
P \left( X < 0.9 \right)
P \left( X > 0.9 \right)
P \left(0.9 < X < 1.5 \right)
A random variable, X, is normally distributed with a mean of 253.4 and a standard deviation of 1.7. Find the following probabilities to four decimal places:
P \left( X < 250.8 \right)
P \left( X > 250.8 \right)
P \left(250.8 < X < 252.9 \right)
A random variable, X, is normally distributed with a mean of - 450.3 and a standard deviation of 75.2. Find the following probabilities to four decimal places:
P \left( X < -294 \right)
P \left( X > -294 \right)
P \left(-469.9 < X < -294 \right)
A random variable, X, is normally distributed with a mean of - 10 and a standard deviation of 98.2. Find the following probabilities to four decimal places:
P \left( X < 178 \right)
P \left( X > 178 \right)
P \left(-44.3 < X < 178 \right)
A random variable, X, is normally distributed with a mean of 50.1 and a standard deviation of 3.2. Determine the probability that X is less than 43.5 or X is greater than 48.5. Round your answer to four decimal places.
A random variable, X, is normally distributed with a mean of - 301.2 and a standard deviation of 53.8. Determine the probability that X is less than - 321.1 or X is greater than - 282.7. Round your answer to four decimal places.
For a set of scores, it was found that the mean score was 61 and the standard deviation was 11. If the distribution of scores is approximately normal, approximate the percentage of scores, to the nearest percent, between:
32 and 76
49 and 52
42 and 62
59 and 60
58 and 84
The heights of 400 netball players were measured and found to fit a normal distribution. The mean height is 149 cm and the standard deviation is 11. Approximate the number of players (to the nearest whole number) that have a height between:
129\text{ cm and }173\text{ cm}
124\text{ cm and }161\text{ cm}
153\text{ cm and }165\text{ cm}
127\text{ cm and }143\text{ cm}
147\text{ cm and }174\text{ cm}
The marks in a class were approximately normally distributed. If the mean was 47 with a standard deviation of 6, approximate (to the nearest whole number) the percentage of students that achieved:
Above the average
Above 42
Below 48
Below 39
Between 46 and 59
The heights of players in a soccer club are approximately normally distributed, with mean height 1.76 m and standard deviation 5 cm. If 700 players are chosen at random, approximately how many players will be taller than 169 cm? Round your answer to the nearest integer.
The times that professional divers can hold their breath are approximately normally distributed with mean 150 seconds and standard deviation 7 seconds. If 400 professional divers are selected at random, approximately how many would be able to hold their breath for longer than 137 seconds? Round your answer to the nearest integer.
The number of biscuits packaged in biscuit boxes is approximately normally distributed with mean 38 and standard deviation 5. If 4000 boxes of biscuits are produced, approximately how many boxes have more than 40 biscuits? Round your answer to the nearest integer.
The exams scores of students are approximately normally distributed with a mean score of 63 and a standard deviation of 8.
What percentage of students scored between 66 and 68? Round your answer to the nearest percent.
There are 450 students who sat the exam. If the passing score is 53, approximately how many students passed? Round your answer to the nearest whole number.
The height of sunflowers is approximately normally distributed, with a mean height of 1.6 m and a standard deviation of 8 cm.
Approximately what percentage of sunflowers are between 1.55 m and 1.60 m tall?Round your answer to the nearest percent.
Approximately what percentage of sunflowers are between 1.69 m and 1.81 m tall? Round your answer to the nearest percent.
If there are 3000 sunflowers in the field, approximately how many are taller than 1.52 m? Round your answer to the nearest integer.
The times that a class of students spent talking or texting on their phones on a particular weekend is approximately normally distributed with mean time 179 minutes and standard deviation 7 minutes. Approximately what percentage of students used their phones for between 186 and 191 minutes on the weekend?
Calculate the percentage of standardised data, to two decimal places, that is:
Calculate the probability, to four decimal places, that a z-score is:
Find the area under the curve, to four decimal places, for each part of the standardised normal curves described below:
Using your calculator, find the value of k to four decimal places for the following probabilities:
Mensa is an organisation that only accepts members who score in the 0.98 quantile or above in an IQ test. Explain what a person must achieve to be accepted into Mensa using percentages.
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
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
A random variable, X, is normally distributed with a mean of 14 and a standard deviation of 3. Find the following probabilities to four decimal places:
The 0.41 quantile
The 87th percentile
A random variable, X, is normally distributed with a mean of - 352 and a standard deviation of 65. Find the following probabilities to four decimal places:
The 0.05 quantile
The 43rd percentile
A random variable, X, is normally distributed with a mean of 25.9 and a standard deviation of 0.7. Find the following probabilities to four decimal places:
The 0.77 quantile
The 23rd percentile
A random variable, X, is normally distributed with a mean of - 0.4 and a standard deviation of 29.7. Find the following probabilities to four decimal places:
The 0.12 quantile
The 72nd 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?
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?
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?
The length of the tail of a domestic cat is normally distributed with a mean of 25 \text{ cm} and a standard deviation of 2.2 \text{ cm}. Use your CAS calculator to answer the following, rounding your answers to three decimal places:
What is the shortest length of a tail in the 70th percentile?
What is the shortest length of a tail in the top 15\%?
What is the shortest length of a tail in the 0.45 quantile?
What is the probability that a cat has a tail length less than 23.5 \text{ cm}?
Suppose that a cat has a tail length below the 80th percentile. What is the probability that their tail length is more than 23.5 \text{ cm}?
The weights of babies born in a hospital in Sydney are considered normally distributed with a mean of 3.3 \text{ kg} and a standard deviation of 0.50 \text{ kg}. Use your CAS calculator to answer the following, rounding your answers to three decimal places:
What is the weight of the smallest baby in the 65th percentile?
What is the weight of the smallest baby in the top 20\%?
What is the weight of the smallest baby in the 0.55 quantile?
What is the probability that a baby weighs less than 3.05 \text{ kg}? Give your answer as a decimal.
Suppose that a baby has a weight below the 70th percentile. What is the probability that their weight is more than 3.05 \text{ kg}?
People in the Dinaric Alps are considered the tallest in the world. The heights of males are considered to be normally distributed with a mean height of 186.0 \text{ cm} and a standard deviation of 6 \text{ cm}. Use your CAS calculator to answer the following, rounding your answers to three decimal places unless otherwise stated:
What is the height of the shortest male in the top 25\%?
What is the height of the tallest male in the 0.95 quantile?
What percentage of males are shorter than 176 \text{ cm}? Round your answer to the nearest tenth of a percent.
What is the probability that a male has a height less than 187 \text{ cm}?
Suppose that a male has a height below the 90th percentile. What is the probability that their height is more than 187 \text{ cm}?
The operating times of phone batteries are approximately normally distributed with a mean of 34 hours and a standard deviation of 4 hours. Answer the following questions using the empirical rule:
What percentage of batteries last between 33 and 38 hours? Round your answer to the nearest percent.
What percentage of batteries last between 28 hours and 41 hours?
Any battery that lasts less than 23 hours is deemed faulty. If a company manufactured 51\,000 batteries, approximately how many batteries would they be able to sell? Round your answer to the nearest integer.