Consider the standard normal curve.
State the size of the area under the curve that is above the mean.
State the size of the area under the curve that is below the mean.
Sketch the standard normal distribution curve and shade in the area where the data values lie in the:
Bottom 10\%.
Top 15\%.
Bottom 67\%.
Top 36\%.
Bottom 42\%.
Represent the following probabilities as areas under a standard normal curve:
P \left(Z\leq 2 \right)
P \left(-2\leq Z\leq 2 \right)
P \left(Z\leq 3 \right)
P \left(-1\leq Z\leq 3 \right)
P \left(-2\leq Z\leq 0 \right)
Consider the standard normal distribution tables which give the area between 0 and a given z-score:
Using the tables, find the area under the normal curve between:
1 standard deviation below the mean and 2 standard deviations above the mean
1.60 and 1.80 standard deviations above the mean
Using the tables from Q4, find the area under the normal curve:
To the left of z = 1.15.
To the right of z = 1.38.
To the left of z = - 1.76.
To the right of z = - 1.16.
Between z = - 1.11 and z = - 1.59.
Between z = 1.51 and z = 1.89.
Using the tables from Q4, find the percentage, to two decimal places, of the data that is:
Less than z = 0.71.
Greater than z = - 1.15.
Less than z = 0.17.
Less than z = 0.84.
Between z = - 1.29 and z = 2.35.
Between z = 0.39 and z = 2.48.
Using the tables from Q4, find the following probabilities for a normally distributed random variable X with the given parameters mean \mu and standard deviation \sigma:
P(3\leq X\leq 7), \mu=5, \sigma=0.8
P(X\geq 20), \mu=4, \sigma=10
P(X\leq 8), \mu=12, \sigma=5
P(X\geq -39), \mu=0, \sigma=30
P(X\lt 36), \mu=20, \sigma=10
P(3\lt X\leq 5), \mu=8, \sigma=2
Consider the probability tables which show the cumulative probabilities of the standard normal distribution:
A data set is normally distributed. Use the probability tables to find the following to two decimal places:
The median.
The 1st quartile.
The upper quartile.
The 84th percentile.
The 23rd percentile.
The 48th percentile.
The 3rd decile.
The 8th decile.
The 4th decile.
Convert these probability statements for the normal random variable X with mean 4 and standard deviation 2 into probability statements on the standard normal random variable Z:
P \left(X\leq 5 \right)
P \left(X\gt 4.5 \right)
P \left(X\leq 2 \right)
P \left(X\geq 1 \right)
P \left(0\leq X\leq 3 \right)
P \left(0.5\leq X\leq 4.5 \right)
Convert these probability statements for the normal random variable X with mean 5 and standard deviation 2 into probability statements on the standard normal random variable Z and then use the empirical rule to find them:
P \left(X \geq 5 \right)
P \left(3\leq X\leq 7 \right)
P \left(X\leq 9 \right)
P \left(X\geq 1 \right)
P \left(-1\leq X\leq 7 \right)
P \left(1\leq X\leq 3 \right)
Use the empirical rule to find the following probabilities for a normally distributed random variable with the given mean \mu and standard deviation \sigma:
P(10\leq X\leq 18), \mu=12, \sigma=2
P(X\geq 42), \mu=37, \sigma=5
P(X\geq 4.5), \mu=4, \sigma=0.25
The average life span of a tortoise is estimated to be approximately 100 years, with a standard deviation of 7 years.
Determine the z-score of a life span of 102 years. Round your answer to two decimal places.
Find P \left(X \gt 102 \right). Round your answer to four decimal places.
Find the probability of a tortoise living beyond 102 years of age.
A sprinter is training for a national competition. She runs 400 \text{ m} in an average time of 75 seconds, with a standard deviation of 6 seconds.
Determine the z-score of a time of 67 seconds. Round your answer to two decimal places.
Find P \left( X \lt 67 \right). Round your answer to four decimal places.
Find the time of the sprinter to run 400\text{ m} that is represented by the probability of 0.0918.
Research scientists have measured the heights of a large number of eucalyptus trees in an area of forest. The average height of these trees is 60\text{ m}, with a standard deviation of 6\text{ m}.
Find the z-score of a height of 68\text{ m}. Round your answer to two decimal places.
Find P \left( X \lt 68 \right). Round your answer to four decimal places.
Find the height of a eucalyptus tree in the forest that is represented by the probability of 0.9082.
The lengths of adult male lizards of a particular species are thought to be normally distributed with a mean of 19.5\text{ cm} and a standard deviation of 4.5\text{ cm}.
Find the z-score of a lizard length of 19.5\text{ cm}.
Find the probability that a randomly chosen adult male lizard of this species will have a length less than 19.5\text{ cm}.
Find the z-score of a lizard length of 15\text{ cm}.
Find the probability that a randomly chosen adult male lizard of this species will have a length between 15\text{ cm} and 19.5\text{ cm}. Round your answer to four decimal places.
A box of breakfast cereal has "contains 500 grams of breakfast cereal" printed on it. The amount of breakfast cereal contained in these boxes is normally distributed with a mean of 512 grams and a standard deviation of 9 grams.
Find the z-score of a box containing 500 grams of cereal, to two decimal places.
Find the probability that a randomly chosen box of this cereal contains less than 500 grams. Round your answer to four decimal places.
In a random sample of 100 boxes of this cereal, find the approximate number of boxes we should expect to contain less than 500 grams.
Pre-bagged packs of bananas are marked as "contains approximately 5\text{ kg}". The mass of the contents of such bags is normally distributed with a mean of 5.01\text{ kg} and a standard deviation of 0.08\text{ kg}.
Find the z-score of a 5\text{ kg} pack of bananas. Round your answer to two decimal places.
Find the probability that a randomly chosen pack of these bananas has a mass of less than 5\text{ kg}. Round your answer to four decimal places.
The scaled results in a national mathematics test are normally distributed with a mean of 65 and a standard deviation of 7.
Find the z-score of a test result of 71. Round your answer to two decimal places.
Find the probability that a randomly selected candidate who sat this test has a scaled result of more than 71. Round your answer to four decimal places.
Find the z-score of a test result of 61. Round your answer to two decimal places.
Find the probability that a randomly selected candidate who sat this test has a scaled result of between 61 and 71. Round your answer to four decimal places.
The scaled result of 51 is two standard deviations below the mean. Find the probability that a randomly selected candidate who sat this test has a scaled result of less than 51. Round your answer to four decimal places.
The mean height of an adult male is 1.78\text{ m}, with a standard deviation of 9\text{ cm}.
Find the z-score of a height of 1.69\text{ m}.
If 700 males are chosen at random, find the approximate number males who are taller than 1.69\text{ m}.
The mean number of biscuits in a box is 35, with a standard deviation of 4.
Find the z-score of a box containing 27 biscuits.
If 4000 boxes of biscuits are produced, find the approximate number of boxes with more than 27 biscuits.
Assume the mean time a male professional diver can hold his breath is 116 seconds, with a standard deviation of 7 seconds.
Find the z-score of a time of 102 seconds.
If 600 male professional divers are selected at random, find the approximate number of divers who can hold their breath for longer than 102 seconds.
The marks of an exam recently completed by a class are normally distributed. The mean mark in the exam was 57, with a standard deviation of 4.
Find the z-score of a mark of 53.
Find the percentage of students who achieved a mark above 53, to the nearest percent.
Find the z-score of a mark of 49.
Find the percentage of students who achieved a mark below 49, to the nearest percent.
Blood pressure is one human characteristic that has a normal distribution. That is, high and low values are unlikely, and average values are more likely.
In the general population, the mean diastolic blood pressure is 83\text{ mmHg} (millimetres of mercury) and the standard deviation is 16\text{ mmHg}.
Find the z-score for a blood pressure of 110\text{ mmHg}, to two decimal places.
Find the percentage of the population with blood pressure greater than 110\text{ mmHg}, to the nearest percent.
Find the z-score for a blood pressure of 42\text{ mmHg}, to two decimal places.
Find the percentage of the population with blood pressure below 42\text{ mmHg}, to the nearest percent.
The time from Laura getting out of bed until her arrival at school is normally distributed with a mean of 36 minutes and a standard deviation of 6 minutes. Laura's arrival at school is classified as being late if it occurs after 9:15 am.
If Laura gets out of bed at 8:29 am, state the number of minutes she has to get to school.
Find the z-score for the time of 46 minutes, to two decimal places.
If Laura gets out of bed at 8:29 am, find the probability that she will arrive late. Round your answer to four decimal places.
A particular vitamin has 42\text{ mg} which is equivalent to 124\% of the recommended daily intake of that vitamin. A 124\text{ ml} box of fruit juice contains approximately 42\text{ mg} of this vitamin. Suppose that the mass of the vitamin in the 124\text{ ml} box of the juice is normally distributed with a mean 42\text{ mg} and standard deviation of 4.5\text{ mg}.
Find 100\% of the recommended daily intake of the vitamin, to the nearest milligram.
Find the z-score of a vitamin mass of 34\text{ mg}, to two decimal places.
Find the probability that a randomly chosen 124\text{ mL} box of this fruit juice contains less than the recommended daily intake of the vitamin, to four decimal places.