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.
x and y represent scores in a normally distributed data set. Consider the following events:
Event 1: Randomly selecting a score in the dataset greater than x.
Event 2: Randomly selecting a score in the dataset greater than y.
Event 3: Randomly selecting a score in the dataset less than x.
-
Event 4: Randomly selecting a score in the dataset less than y.
For each of the following, state the event that is the most likely to occur:
The z-score of x is 2 and the z-score y is 3.
The z-score of x is -1 and the z-score y is -2.
The z-score of x is - 0.96 and the z-score y is - 0.21.
For each of the following, state the event that is the least likely to occur:
The z-score of x is 2.03 and the z-score y is 2.77.
The z-score of x is 1 and the z-score y is -2.
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 finish times for a half-marathon are approximately normally distributed. Two runners completed the half-marathon in 158 and 106 minutes respectively. A finish time of 158 minutes has a z-score of 2 and a finish time of 106 minutes has a z-score of - 1.
State the event that is more likely to occur:
Event 1: A randomly chosen runner completed the half-marathon in less than 106 minutes.
-
Event 2: A randomly chosen runner completed the half-marathon in more than 158 minutes.
State the event that is more likely to occur:
-
Event 3: A randomly chosen runner completed the half-marathon in more than 106 minutes.
Event 4: A randomly chosen runner completed the half-marathon in less than 158 minutes.
The amount of time spent waiting in the reception area at a doctor's office is approximately normally distributed. The z-scores of the waiting times of four patients are represented by the letters K, L, M, and N:
| \text{Patient} | K | L | M | N |
|---|---|---|---|---|
| z-\text{score} | -2.91 | 1.84 | -2.15 | 1.48 |
Which event is the least likely?
Waiting longer than L.
Waiting longer than M.
Waiting longer than N.
Waiting longer than K.
State the event that is more likely to occur:
-
Event 1: Waiting longer than M but less than N.
-
Event 2: Waiting longer than K but less than L.
State the event that is more likely to occur:
-
Event 3: Waiting less than M or longer than N.
-
Event 4: Waiting less than K or longer than L.
The class exam results for a particular subject is approximately normally distributed. The names of four different students along with the z-scores for their exam results are given in the table:
| \text{Student} | \text{Ben} | \text{Laura} | \text{Neville} | \text{Danielle} |
|---|---|---|---|---|
| z-\text{score} | -1.01 | 4.08 | -1.67 | 4.27 |
It is least likely to randomly select a student that scored higher in the exam than which student?
Ben
Laura
Neville
Danielle
State the event that is more likely to occur:
Event 1: Randomly selecting a student who scored in the exam between Ben and Laura
-
Event 2: Randomly selecting a student who scored in the exam between Neville and Danielle
State the event that is more likely to occur:
-
Event 3: Randomly selecting a student who scored less than Ben or greater than Laura.
-
Event 4: Randomly selecting a student who scored less than Neville or greater than Danielle.
The heights of sunflowers at a nursery are approximately normally distributed with mean 185\text{ cm} and standard deviation 29\text{ cm}. The heights and their z-scores are given in the table:
| \text{Height (cm)} | 211.1 | 77.7 |
|---|---|---|
| z-\text{score} | 0.9 | -3.7 |
State the event that is more likely to occur:
-
Event 1: Randomly selecting a sunflower from the nursery that's taller than 77.7\text{ cm}.
-
Event 2: Randomly selecting a sunflower from the nursery that's shorter than 211.1\text{ cm}.
The number of babies born in a country each day is approximately normally distributed with mean 153 and standard deviation 28. The number of babies born on two consecutive days along with their \\ z-scores are shown in the table:
| \text{Number of newborns} | 265 | 69 |
|---|---|---|
| z-\text{score} | 4 | -3 |
State the event that is more likely to occur:
-
Event 1: The number of babies born the following day is greater than 265.
Event 2: The number of babies born the following day is less than 69.
A data set is approximately normally distributed with mean 5 and standard deviation 3. Both 17 and 17.39 are scores from the data set.
Find the value of the z-score that corresponds to a score of:
17
17.39
State the event that is more likely to occur:
Event 1: Randomly selecting a score in the data set greater than 17.
-
Event 2: Randomly selecting a score in the data set greater than 17.39.
State the event that is more likely to occur:
Event 3: Randomly selecting a score in the data set less than 17.39.
-
Event 4: Randomly selecting a score in the data set less than 17.
A data set is approximately normally distributed with mean - 6 and standard deviation 3. Both - 18.45 and - 18 are scores from the data set.
Find the value of the z-score that corresponds to a score of:
- 18.45
- 18
State the event that is less likely to occur:
-
Event 1: Randomly selecting a score in the data set less than - 18.
-
Event 2: Randomly selecting a score in the data set less than - 18.45.
State the event that is less likely to occur:
-
Event 3: Randomly selecting a score in the data set greater than - 18.
-
Event 4: Randomly selecting a score in the data set greater than - 18.45.
A data set is approximately normally distributed with mean 27 and standard deviation 4.
Complete the following table:
| \text{Scores} | 19.88 | 29.76 | 22.88 | 29.28 |
|---|---|---|---|---|
| z-\text{scores} | -1.78 | 0.69 |
Which event is most likely? Randomly selecting a score in the data set more than:
22.88
29.28
19.88
29.76
State the event that is less likely to occur:
-
Event 1: Randomly selecting a score in the data set between 22.88 and 29.28
-
Event 2: Randomly selecting a score in the data set between 19.88 and 29.76
State the event that is less likely to occur:
Event 3: Randomly selecting a score in the data set that is less than 19.88 or greater than 29.76
Event 4: Randomly selecting a score in the data set that is less than 22.88 or greater than 29.28
The lengths of Swordfish are approximately normally distributed with mean 240 \text{ cm} and standard deviation 20 \text{ cm} .
Complete the following table:
| \text{Lengths (cm)} | 174.2 | 249.2 | 166.4 | 251.6 |
|---|---|---|---|---|
| z-\text{scores} | -3.29 | 0.46 |
Which event is least likely? Capturing a Swordfish more than:
174.2\text{ cm}
251.6\text{ cm}
249.2\text{ cm}
166.4\text{ cm}
State the event that is less likely to occur:
-
Event 1: Capturing a swordfish that is between 174.2\text{ cm} and 249.2\text{ cm}
Event 2: Capturing a swordfish that is between 166.4\text{ cm} and 251.6\text{ cm}
State the event that is less likely to occur:
-
Event 3: Capturing a swordfish that is less than 174.2\text{ cm} or greater than 249.2\text{ cm}
Event 4: Capturing a swordfish that is less than 166.4\text{ cm} or greater than 251.6\text{ cm}
The heights of office buildings in a particular city is approximately normally distributed with mean 60\text{ m} and standard deviation 2\text{ m}. Both 57.48\text{ m} and 57.24\text{ m} are the heights of two office buildings.
Find the value of the z-score that corresponds to a height of:
57.48\text{ m}
57.24\text{ m}
State the event that is less likely to occur:
-
Event 1: A randomly chosen building is taller than 57.48\text{ m}.
Event 2: A randomly chosen building is shorter than 57.24\text{ m}.
State the event that is less likely to occur:
Event 3: A randomly chosen building is taller than 57.48\text{ m}.
-
Event 4: A randomly chosen building is shorter than 57.24\text{ m}.
The profit a restaurant earns each day is approximately normally distributed with mean \$790 and standard deviation \$40. In the last two days, the restaurant profited \$860.80 and \$736.00 respectively.
Find the z-score that corresponds to a profit of:
\$860.80
\$736.00
State the event that is less likely to occur:
Event 1: That the restaurant profits more than \$860.80 today.
-
Event 2: That the restaurant profits less than \$736.00 today.
State the event that is more likely to occur:
-
Event 3: That the restaurant profits less than \$860.80 today.
-
Event 4: That the restaurant profits more than \$736.00 today.
A data set is approximately normally distributed with standard deviation 3. Both 14.12 and 14.99 are scores from the data set. The value of 14.12 has a z-score of 2.04.
Find the mean of the data set.
Find the z-score that corresponds to a score of 14.99.
State the event that is less likely to occur:
-
Event 1: Randomly selecting a score in the dataset greater than 14.12.
Event 2: Randomly selecting a score in the dataset greater than 14.99.
State the event that is less likely to occur:
Event 3: Randomly selecting a score in the dataset less than 14.99.
-
Event 4: Randomly selecting a score in the dataset less than 14.12.
The time it takes for an athlete to complete the bicycle stage of a triathlon is approximately normally distributed with standard deviation 4\text{ min}. The elapsed times for two athletes at each stage is recorded in the table:
| \text{Athlete } 1 | \text{Swim (min)} | \text{Bicycle (min)} | \text{Run (min)} |
|---|---|---|---|
| \text{Hannah} | 46 | 133 | 198 |
| \text{Elizabeth} | 48 | 137 | 206 |
Find the time it takes for Hannah to complete the bicycle stage of the race.
Find the time it takes for Elizabeth to complete the bicycle stage of the race.
The time it takes Hannah to complete the bicycle stage has a z-score of 1.25. Find the average time it takes for an athlete to complete the bicycle stage of the triathlon.
Find the z-score for the time it takes Elizabeth to complete the bicycle stage of the triathlon.
State the event that is less likely to occur:
Event 1: Randomly choosing an athlete that completed the bicycle stage slower than 87\text{ min}.
-
Event 2: Randomly choosing an athlete that completed the bicycle stage slower than 89\text{ min}.
State the event that is less likely to occur:
-
Event 3: Randomly choosing an athlete that completed the bicycle stage quicker than 89 mins.
Event 3: Randomly choosing an athlete that completed the bicycle stage quicker than 87 mins.
The amount of soft drink in a can is normally distributed with standard deviation 0.2\text{ ml}. Two cans contain 374.82\text{ ml} and 374.92\text{ ml} of soft drink. A can with 374.82\text{ ml} of soft drink has a \\ z-score of - 0.9.
Find the average volume of soft drink in a can.
Find the value of the z-score for a can with 374.92\text{ ml} of soft drink.
State the event that is more likely to occur:
Event 1: Purchasing a can of soft drink with less than 374.82\text{ ml}.
Event 2: Purchasing a can of soft drink with less than 374.92\text{ ml}.
State the event that is more likely to occur:
Event 3: Purchasing a can of soft drink with more than 374.92\text{ ml}.
Event 4: Purchasing a can of soft drink with more than 374.82\text{ ml}.
The number of coffees a cafe sells in a weekday is approximately normally distributed with a standard deviation of 9 coffees.
On Monday, the cafe sells 266 cups of coffee which has a z-score of 1. Find the average number of coffees the cafe sells on a weekday.
Complete the following table:
| Monday | Tuesday | Wednesday | Thursday | |
|---|---|---|---|---|
| \text{Coffees} | 266 | 284 | 221 | 248 |
| z-\text{scores} | 1 | 3 |
State the event that is less likely to occur:
-
Event 1: The cafe will sell between 248 and 266 coffees on the Friday.
Event 2: The cafe will sell between 221 and 284 coffees on the Friday.
State the event that is the least likely to occur:
Event 3: The cafe will sell more than 284 coffees on Friday.
-
Event 4: The cafe will sell more than 266 coffees on Friday.
-
Event 5: The cafe will sell more than 221 coffees on Friday.
-
Event 6: The cafe will sell more than 248 coffees on Friday.
The times at which individuals visit a cafe are recorded every hour. The data set is approximately normally distributed with the busiest time at 9:00 am and a standard deviation of 15 minutes.
Complete the following table:
| \text{Times} | \text{9:15 am} | \text{9:30 am} | \text{9:45 am} |
|---|---|---|---|
| z-\text{scores} |
The cafe owner has a limited number of cakes. To maximise their chance of sales, state the 15 minute interval between 9 am and 10 am that they should have the cakes available.
The owner of a cafe records the arrival times of every customer each morning. The data set is approximately normally distributed with the busiest time at 9:00 am and a standard deviation of 20 minutes.
Complete the following table:
| \text{Times} | \text{8:00 am} | \text{8:20 am} | \text{8:40 am} |
|---|---|---|---|
| z-\text{scores} |
The cafe owner has a friend who can help for 20 minutes in the morning. State the range of the busiest time period, between 8 am and 9 am, for her friend to help.
The arrival time of a particular train is normally distributed with an expected arrival of
6:00 pm and a standard deviation of 10 minutes.
Complete the following table:
| \text{Times} | \text{5:40 pm} | \text{5:50 pm} | \text{6:00 pm} | \text{6:10 pm} | \text{6:20 pm} |
|---|---|---|---|---|---|
| z-\text{scores} |
The train scheduled to arrive at 6:00 pm. State the 20-minute time interval in which the train will most likely arrive.
Kate owns a farm with pet chickens. The number of eggs the chickens lay on a given day is approximately normally distributed with standard deviation 9.
On Monday, the chickens laid 92 eggs which has a z-score of 1. Find the average number of eggs the chickens lay on a given day.
Complete the table by finding the rest of the z-scores.
| Monday | Tuesday | Wednesday | Thursday | |
|---|---|---|---|---|
| \text{Eggs} | 92 | 101 | 65 | 56 |
| z-\text{scores} | 1 | 2 |
Next week Kate wants to bake cakes for a work function using the eggs laid that will be laid tomorrow. Find the number of eggs that Kate will most likely to obtain.
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.