Normal distribution + bell curve (+inv normal): notes and practice questions
- A continuous, symmetric, bell-shaped curve representing data distributed around a mean () with standard deviation ().
- Probability is calculated using the z-score:
- Inverse normal finds the value given a cumulative probability.
- Key properties:
- 68% of data lies within 1, 95% within 2, 99.7% within 3.
How it is examined
A question that asks for or from a probability belongs at SL 4.12. A labelled sketch of the curve with the region shaded earns marks. Paper 2, 5 to 8 marks.
Nothing. The booklet has no entry at SL 4.9 at all: no normal probability density function, no distribution table. is in the guide's notation list, not the booklet, and the GDC does the rest.
- The normal distribution and curve.
- Properties of the normal distribution.
- Diagrammatic representation.
- Normal probability calculations.
Linking questions
- Links to other subjects: normally distributed real-life measurements (sciences, psychology).
Practice questions
23 questions · 14 medium · 9 hardQuestion 1
MediumPaper 2 · calculator5 marksThe weights of apples harvested from a particular orchard are normally distributed with a mean of 180 grams and a standard deviation of grams. The interquartile range (IQR) of the apple weights is 12 grams.
Find the value of .
Recall that for a normal distribution, the interquartile range (IQR) is the difference between the 75th percentile () and the 25th percentile (). You will need to find the z-score corresponding to these percentiles.
Question 2
HardPaper 2 · calculator5 marksA factory produces two types of light bulbs: standard and long-life. The lifespan of the bulbs, in hours, can be modelled as normal distributions with the following parameters.
| Bulb type | Mean | Standard deviation |
|---|---|---|
| Standard | 800 h | 50 h |
| Long-life | 1000 h | 100 h |
(a) Find the percentage of standard bulbs that have a lifespan of less than 700 hours.
(b) The factory produces a large number of bulbs, of which 60% are standard bulbs. Both types of bulbs are produced and randomly mixed together for packaging. A quality control process identifies and removes all bulbs with a lifespan of less than 700 hours. An inspector randomly selects a bulb from this removed group. Find the probability that it is a standard bulb.
Use the normal distribution CDF with mean 800 and standard deviation 50 to find the probability of a lifespan less than 700 hours, then express it as a percentage.
This is a conditional probability problem. Let S be the event that a bulb is standard, and L be the event that its lifespan is less than 700 hours. You need to find P(S|L). Use the formula for conditional probability or Bayes' theorem. You will need to calculate the probability of a long-life bulb having a lifespan less than 700 hours first.
Question 3
MediumPaper 2 · calculator6 marks(a) The lifespan of a certain type of rechargeable battery, in hours, can be modelled by a normal distribution with a mean of 1500 hours and a standard deviation of 50 hours. A battery is deemed faulty and rejected if its lifespan is less than 1425 hours.
Find the probability that a randomly selected battery is rejected.
(b) Estimate the number of batteries that will be rejected from a random sample of 200 batteries.
(c) Given that a battery is not rejected, find the probability that it has a lifespan greater than 1575 hours.
Use the normal cumulative distribution function (CDF) to find the probability that the lifespan is less than the rejection threshold. Remember to use the given mean and standard deviation.
Multiply the probability of a single battery being rejected by the total number of batteries in the sample.
This is a conditional probability problem. You need to find . Remember the formula for conditional probability: .
Question 4
HardPaper 2 · calculator16 marks(a) The random variable follows a normal distribution with mean and standard deviation .
Find .
(b) The diameters of ball bearings produced by a factory, in mm, are normally distributed with mean and standard deviation . The ball bearings are categorized as defective, standard, large, or premium, according to their diameter. The following table shows the probability a ball bearing is classified into each category.
| Category | Probability |
|---|---|
| Defective | 0.03 |
| Standard | 0.65 |
| Large | 0.25 |
| Premium | 0.07 |
The maximum diameter of a defective ball bearing is 14.8 mm.
The minimum diameter of a premium ball bearing is 16.5 mm.
Find the value of and of .
(c) The factory rejects all defective ball bearings. The remaining ball bearings are sold.
Find the probability that a ball bearing chosen at random from those sold is categorized as
(i) standard;
(ii) large;
(iii) premium.
(d) The selling prices of the different categories of ball bearings at this factory are shown in the following table:
| Category | Selling Price ($) |
|---|---|
| Standard | 1.50 |
| Large | 1.80 |
| Premium | 2.50 |
The factory incurs a fixed cost of $300 for the production run and assumes it will sell the accepted ball bearings in exactly the same proportion as calculated in part (c).
According to this model, find the minimum number of accepted ball bearings that must be sold so that the net profit for the factory is at least $550.
Recall that for a normal distribution, you can standardize the variable to a standard normal variable using the formula . Then use your GDC to find the probability.
Use the given probabilities and boundary values to find the corresponding z-scores. Then, set up two simultaneous equations involving and and solve them.
This is a conditional probability problem. The new sample space consists only of non-defective ball bearings.
Remember to use the new sample space (non-defective ball bearings) for this conditional probability.
The denominator for the conditional probability remains the probability of a non-defective ball bearing.
First, calculate the expected revenue per accepted ball bearing using the probabilities from part (c) and the selling prices. Then, set up an inequality for the total profit.
Question 5
MediumPaper 2 · calculator6 marks(a) A beverage company fills bottles with juice. The volume of juice, in millilitres (ml), can be modelled by a normal distribution with a mean of 750 ml and a standard deviation of 2.5 ml. A bottle is considered underfilled and rejected if its volume is less than 746 ml.
Find the probability that a randomly selected bottle is rejected.
(b) Estimate the number of bottles that will be rejected from a random sample of 200 bottles.
(c) Given that a bottle is not rejected, find the probability that its volume is greater than 753 ml.
Recall the properties of the normal distribution. You need to calculate the probability of the volume being less than a certain value. Use your GDC's normal cumulative distribution function (normalcdf).
Multiply the probability of rejection by the total number of bottles in the sample.
This is a conditional probability problem. Remember that 'not rejected' means the volume is 746 ml or more. You need to find .
Question 6
HardPaper 2 · calculator16 marks(a) An electronics factory produces two types of resistors: Type A and Type B.
The resistance, (in Ohms), of Type A resistors is normally distributed with a mean of 120 Ohms and a standard deviation of 5 Ohms.
Find the probability that a randomly selected Type A resistor has a resistance less than 115 Ohms.
(b) In a random selection of 10 Type A resistors, find the probability that exactly 3 have a resistance less than 115 Ohms.
(c) The resistance, (in Ohms), of Type B resistors is normally distributed with a mean of 135 Ohms and a standard deviation of 7 Ohms.
Each day, 70% of the resistors produced are Type A, and 30% are Type B.
On a particular day, a resistor is randomly selected from all those produced at the factory.
Let represent 'Type A resistor' and represent 'Type B resistor'.
(i) Find the probability that the randomly selected resistor has a resistance less than 115 Ohms.
(ii) Given that a randomly selected resistor has a resistance less than 115 Ohms, find the probability that it is a Type A resistor.
(d) The machine that makes the Type A resistors is adjusted so that the mean resistance of the Type A resistors remains the same, but their standard deviation changes to Ohms. The machine that makes the Type B resistors is not adjusted. The probability that the resistance of a randomly selected resistor from these machines is now less than 115 Ohms is 0.18.
Find the value of .
Use the normal cumulative distribution function (CDF) on your GDC. Remember to input the lower bound, upper bound, mean, and standard deviation.
This is a binomial probability problem. Identify the number of trials, the number of successes, and the probability of success from part (a).
You need to consider both types of resistors. Calculate the probability for Type B resistors first, then use the law of total probability, taking into account the proportions of each type.
This is a conditional probability problem, often solved using Bayes' theorem. You need the probability of a Type A resistor having low resistance and the overall probability of a low resistance resistor.
Set up an equation for the new total probability, similar to part (c.i). You'll need to solve for the new probability , then use the inverse normal function to find the z-score, and finally calculate the new standard deviation .
Question 7
MediumPaper 2 · calculator15 marksThe delivery times, minutes, for packages from a logistics hub to a regional distribution center can be modelled by a normal distribution with a mean of 120 minutes and a standard deviation of minutes.
Given that 3% of the delivery times are longer than 135 minutes, find the value of .
Find the probability that a randomly selected package will have a delivery time of more than 130 minutes.
Given that a package delivery takes longer than 130 minutes, find the probability that it takes less than 135 minutes.
On a particular day, there are 80 packages scheduled for delivery from the hub.
Find the expected number of packages that will have a delivery time of more than 130 minutes.
Find the probability that more than 8 of the packages on this particular day will have a delivery time of more than 130 minutes.
Use the inverse normal distribution function on your GDC to find the z-score corresponding to the given percentile. Remember that if 3% are longer than 135 minutes, then 97% are shorter than 135 minutes.
Use the normal cumulative distribution function (CDF) on your GDC. Remember to use the standard deviation found in part (a).
This is a conditional probability problem. Recall the formula . Here, event A is 'delivery takes less than 135 minutes' and event B is 'delivery takes longer than 130 minutes'.
This involves a binomial distribution. The expected value for a binomial distribution is given by , where is the number of trials and is the probability of success for a single trial (from part (b) ).
Use the binomial cumulative distribution function (CDF) on your GDC. Remember that 'more than 8' means , which can be calculated as .
Question 8
HardPaper 2 · calculator18 marks(a) A new automated coffee machine is programmed to dispense coffee. The volume of coffee dispensed, ml, is normally distributed with a mean of 200 ml and a standard deviation of ml.
On 15% of occasions, the machine dispenses more than 210 ml of coffee.
Find the value of .
(b) On a randomly selected occasion, find the probability that the machine dispenses more than 205 ml of coffee.
(c) The machine is considered to have 'over-filled' a cup if it dispenses more than 215 ml of coffee. Seven customers order coffee. Assume the volume dispensed for each customer is independent.
Find the probability that at least one of these seven coffees is over-filled.
(d) Given that at least one of the seven coffees is over-filled, find the probability that exactly two of them are over-filled.
(e) The café serves 25 customers in an hour. The machine requires maintenance if it over-fills more than 3 coffees during that hour. So far, 18 customers have been served, and the machine has over-filled 2 coffees.
Find the probability that the machine will NOT require maintenance by the end of the hour.
For a normal distribution, you can use the inverse normal function on your GDC to find the z-score corresponding to a given probability. Remember the formula for the z-score: .
Use the standard deviation found in part (a) and the normal distribution function on your GDC to find the probability.
First, calculate the probability of a single coffee being over-filled. Then, consider a binomial distribution for the number of over-filled coffees among the seven customers. 'At least one' often implies using the complementary probability.
This is a conditional probability problem. Remember the formula . Here, event A is 'exactly two over-filled' and event B is 'at least one over-filled'. What is the intersection of these two events?
Determine how many more customers will be served and how many more over-fills are allowed to avoid maintenance. Then, use the binomial distribution for the remaining trials.
Question 9
MediumPaper 2 · calculator15 marks(a) The lifespan of a new type of battery, hours, can be modelled by a normal distribution with a mean of 1200 hours and a standard deviation of hours.
Given that 5% of the batteries last longer than 1280 hours, find the value of .
(b) Find the probability that a randomly selected battery will have a lifespan of less than 1150 hours.
(c) Given that a battery lasts longer than 1150 hours, find the probability that it lasts less than 1250 hours.
(d) A batch of 500 batteries is produced. Find the expected number of batteries that will have a lifespan of less than 1150 hours.
(e) Find the probability that more than 85 of the batteries in this batch will have a lifespan of less than 1150 hours.
For part (a), use the inverse normal function on your GDC to find the z-score corresponding to the given percentile. Then, use the z-score formula to solve for . Remember that 5% lasting longer means 95% last less than that value.
For part (b), use the normal cumulative distribution function (CDF) on your GDC with the mean and standard deviation found in part (a). You need to find .
For part (c), this is a conditional probability problem. Recall the formula . Here, A is 'lasts less than 1250 hours' and B is 'lasts longer than 1150 hours'. So you need to find and .
For part (d), this involves a binomial distribution. The number of trials is the batch size, and the probability of success is the probability calculated in part (b). The expected number of successes in a binomial distribution is given by .
For part (e), you need to calculate for the binomial distribution . Remember that , and your GDC can compute using binomial CDF.
Question 10
HardPaper 2 · calculator16 marksThe lifespan, (in hours), of a certain type of LED light bulb is modelled by a normal distribution with mean and standard deviation .
It is known that and .
Find the probability that a randomly selected light bulb has a lifespan between hours and hours.
Find the value of and the value of .
A manufacturer tests a batch of randomly selected light bulbs. Any bulb with a lifespan greater than hours is considered a 'long-life' bulb. Lifespans of bulbs are independent of each other.
Find the probability that exactly bulbs in the batch are 'long-life' bulbs.
Given that fewer than bulbs are 'long-life' bulbs, find the probability that exactly bulbs are 'long-life' bulbs.
In another factory, a different type of LED light bulb is produced. The lifespan of these bulbs, (in hours), is normally distributed with a mean of hours. The interquartile range (IQR) for these bulbs is hours.
Find the value of the standard deviation, , for this type of bulb.
Recall that the sum of probabilities for all possible outcomes in a continuous distribution is 1. Consider the regions defined by the given probabilities.
Use the inverse normal function to find the z-scores corresponding to the given probabilities. Then set up a system of two linear equations using the formula and solve for and .
This scenario involves a fixed number of trials (bulbs), two possible outcomes ('long-life' or not), and independent trials. This suggests a binomial distribution.
This is a conditional probability problem. Remember the formula . Here, event A is 'exactly 25 bulbs are long-life' and event B is 'fewer than 30 bulbs are long-life'.
The interquartile range is the difference between the upper quartile () and the lower quartile (). For a normal distribution, corresponds to the 25th percentile and to the 75th percentile. Use the inverse normal function to find the z-scores for these percentiles.
Question 11
MediumPaper 2 · calculator5 marksA beverage company uses automated machinery to fill bottles with orange juice. The volume of juice, ml, dispensed into each bottle is normally distributed with a mean of 500 ml and a standard deviation of .
The interquartile range of the volumes is 2.4 ml.
Find the value of .
Recall that the interquartile range (IQR) is the difference between the third quartile () and the first quartile (). For a normal distribution, and correspond to specific z-scores that can be found using the inverse normal function.
Question 12
HardPaper 2 · calculator16 marks(a) The resistance, R ohms, of resistors produced by a factory is normally distributed with a mean of 100 ohms and a standard deviation of 3.5 ohms.
Find the probability that a randomly selected resistor has a resistance less than 98 ohms.
(b) In a random sample of 15 resistors, find the probability that exactly 4 of them have a resistance less than 98 ohms.
(c.i) The capacitance, C microfarads, of capacitors produced by the same factory is normally distributed with a mean of 50 F and a standard deviation of 2.8 F. Each day, 70% of the components produced are resistors and 30% are capacitors.
Find the probability that a randomly selected component has a value less than its respective threshold (i.e., less than 98 ohms for a resistor or less than 47 F for a capacitor).
(c.ii) Given that a randomly selected component has a value less than its respective threshold, find the probability that it is a resistor.
(d) The resistor manufacturing process is adjusted so that the mean resistance remains 100 ohms but its standard deviation changes to ohms. The capacitor manufacturing process is not adjusted. The probability that a randomly selected component from these machines has a value less than its respective threshold is now 0.160.
Find the value of .
Use the normal cumulative distribution function (CDF) on your GDC. Remember to input the lower bound, upper bound, mean, and standard deviation.
This is a binomial probability problem. Identify the number of trials (n), the number of successes (k), and the probability of success (p) from part (a).
First, find the probability that a capacitor has a capacitance less than 47 F. Then, use the law of total probability, considering the proportion of resistors and capacitors produced.
This is a conditional probability problem. Use Bayes' theorem: P(A|B) = P(A and B) / P(B).
Work backwards. Use the new total probability and the unchanged capacitor probability to find the new probability for resistors. Then use the inverse normal function to find the z-score, and finally calculate the new standard deviation.
Question 13
MediumPaper 2 · calculator7 marksLiam and Chloe are participating in an archery tournament.
The scores, points, achieved by Liam on a target can be modelled by a normal distribution with mean 85 and standard deviation 4.
The scores, points, achieved by Chloe on a target can be modelled by a normal distribution with mean 88 and standard deviation 3.
In the first round of the tournament, each competitor takes six shots. To qualify for the next round, a competitor must achieve at least one score of 90 points or greater in the first round.
Find the probability that only one of Liam or Chloe qualifies for the next round of the tournament.
First, calculate the probability of a single successful shot for each competitor using the normal distribution. Then, use the binomial distribution to find the probability that each competitor qualifies for the round. Finally, combine these probabilities to find the chance that only one qualifies.
Question 14
HardPaper 2 · calculator10 marks(a) The battery life, hours, of a new smartphone model is normally distributed with a mean of hours and a standard deviation of hours.
Find the probability that a randomly selected smartphone will have a battery life of more than hours.
(b) The manufacturer offers a free replacement for any smartphone whose battery life is less than hours. Find the percentage of smartphones that will need to be replaced.
(c) If two smartphones are purchased, find the probability that both will have a battery life between hours and hours.
Remember that for a normal distribution, you can use a GDC or standardize the variable to find probabilities. For , you can calculate .
To find the percentage, calculate the probability as a decimal and then multiply by . For , you can directly use the normal CDF function on your GDC.
First, find the probability that one smartphone's battery life is within the given range. Then, consider how the probabilities of two independent events are combined.
Question 15
MediumPaper 2 · calculator8 marks(a) The volume, ml, of liquid in bottles filled by a machine can be modelled by a normal distribution with mean ml and standard deviation ml.
A bottle is selected at random.
Find the probability that it contains more than ml.
(b) According to this model, of the bottles contain between ml and ml.
Find the probability that a randomly selected bottle contains less than ml.
(c) Find the value of .
(d) A quality control inspector randomly selects bottles.
Find the probability that exactly of these bottles contain less than ml.
Use your GDC to find the probability for a normal distribution. Remember to use the correct parameters for mean and standard deviation.
Consider the total probability under the curve and the probabilities you already know. The total area under the probability density function is 1.
You need to use the inverse normal function on your GDC. Ensure you use the cumulative probability for .
This is a binomial probability scenario. Identify the number of trials, the number of successes, and the probability of success from previous parts.
Question 16
HardPaper 2 · calculator14 marksThe mass, grams, of mangoes grown in an orchard can be modelled by a normal distribution with mean and standard deviation . The masses of all mangoes are independent of each other.
of the mangoes have a mass of less than grams, while of the mangoes have a mass of more than grams.
Find the value of and the value of .
Hence, find the probability that a mango chosen at random has a mass of between and grams.
The farmer also grows avocados. Records show that of the avocados grown are classified as large.
The probability of an avocado being classified as large is independent of any other avocado.
On a particular day, avocados are randomly selected.
Find the probability that at least of these avocados are classified as large.
Given that at least of these avocados are classified as large, find the probability that more than are not classified as large.
Use the inverse normal function on your calculator to find the z-scores for the given probabilities, then set up a system of equations.
Use the normal cumulative distribution function (normal CDF) on your calculator with the mean and standard deviation you just found.
This is a binomial distribution problem. You need to find the probability of 15 or more successes out of 24 trials.
Use the conditional probability formula: . Think carefully about what 'more than 4 are not large' means in terms of the number of large avocados.
Question 17
MediumPaper 2 · calculator13 marks(a) A barista prepares a special coffee drink. The time, in minutes, taken to prepare this drink is modelled by a normal distribution with a mean of minutes and a standard deviation of minutes.
Find the probability that on a randomly selected order, the barista takes longer than minutes to prepare the drink.
(b) Find the probability that on a randomly selected order, the barista takes between and minutes to prepare the drink.
(c) If the probability that the barista takes longer than minutes to prepare the drink is , find the value of .
(d) Over a month, the barista prepares this special coffee drink for customers. Find the number of customers for whom the barista can expect to prepare the drink in under minutes.
Use the normal distribution cumulative distribution function (CDF) or a GDC to find the probability . Remember that .
To find , you can calculate using the normal CDF or a GDC.
You are given . This means . Use the inverse normal function on your GDC.
First, find the probability . Then, multiply this probability by the total number of customers to find the expected number.
Question 18
HardPaper 2 · calculator7 marksThe masses of a certain type of melon are normally distributed with mean and standard deviation .
The probability that a randomly chosen melon has a mass greater than is .
If a melon with a mass greater than is chosen at random, the probability that its mass is greater than is .
Find the value of and the value of .
Use the conditional probability formula to find the overall probability that a melon has a mass greater than . Then, standardise both values to set up a system of two equations.
Question 19
MediumPaper 2 · calculator11 marksThe weight of bags of coffee beans produced by a certain machine is modelled by a normal distribution with mean grams and standard deviation grams.
It is known that and , where is the weight of a randomly selected bag of coffee beans.
Find the value of and the value of .
Find the range of weights, symmetric about the mean, such that the probability of a randomly selected bag falling within this range is .
For a normal distribution , you can standardize values using . Use the inverse normal function to find the -scores corresponding to the given probabilities. This will lead to a system of two linear equations in terms of and .
If the range is symmetric about the mean, say , then . Use the symmetry of the normal distribution to find the probability , then find the corresponding -score and solve for .
Question 20
MediumPaper 2 · calculator7 marks(a) A coffee roasting company packages its premium blend coffee into bags. The weight of coffee in these bags is assumed to be normally distributed. Bags are rejected if their weight is less than grams or more than grams.
It is found that of the bags are rejected for being too light, and of the bags are rejected for being too heavy. Find the mean and the standard deviation of the weight of the coffee bags. Give your answers correct to significant figures.
For a normal distribution, you can use the inverse normal function (or Z-tables) to find the Z-scores corresponding to the given probabilities. Remember that is equivalent to . Once you have two Z-scores and their corresponding X-values, you can set up a system of two linear equations in terms of and and solve them simultaneously.
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Where marks are lost
- Rounding an intermediate value and then using it.
- Answering to the wrong accuracy. Two significant figures, or six, where the rule says exactly or three.
- Writing the answer and nothing else.
