7.03 Normal distribution
State whether the following curves represent the shape of normally distributed data:
State whether the following sets of data are normally distributed:
Consider the following normal distribution:
Find:
The mean
The median
The mode
In a normal distribution, state the percentage of data that lies:
Above the mean
Below the mean
A data set has a mean of 80 and a standard deviation of 3. What score is 3 standard deviations above the mean?
The diagram below shows the distribution of females heights in a population:
If 34\% of females lie between 157 cm and 163 cm, what percentage of females lie between 163 \text{ cm} and 169 \text{ cm}.
What length is one standard deviation equal to?
The results from an exam were normally distributed. The mean score was 65, with a standard deviation of 9. The points marked on the horizontal axis are separated by one standard deviation:
What is the value of x on the graph?
What is the value of y on the graph?
In a male population, the mean height was 175 \text{ cm}, with a standard deviation of 7 \text{ cm}.
How tall is Bob who is 4 standard deviations above the mean?
How tall is John who is 2 standard deviations below the mean?
A sample of professional basketball players is normally distributed and gives the mean height as 199 cm with a standard deviation of 10 \text{ cm}.
How tall is a basketball player who is 3 standard deviations above the mean?
How tall is a basketball player who is 1.5 standard deviations below the mean?
Assume the mass of sumo wrestlers is normally distributed, with a mean mass of 158 kg and a standard deviation of 10 kg.
What percentage of wrestler would weigh more than 158 kg?
A sumo wrestler's weight is one standard deviation below the mean. How much does he weigh?
The following figure shows the approximate percentage of scores lying within various standard deviations from the mean of a normal distribution:
A set of scores is found to follow such a distribution, where the mean score is 92 and standard deviation is 20. Find the percentage of scores between:
72 \text{ and } 112
52 \text{ and } 132
32 \text{ and } 152
92 \text{ and } 112
112 \text{ and } 132
Use the empirical rule to calculate the percentage of scores in a normal distribution that lie between:
For each normal distribution shown below, each unit on the horizontal axis indicates 1 standard deviation. Use the empirical rule to find the percentage of scores that lie in the shaded region:
The heights of 600 boys are found to approximately follow such a distribution, with a mean height of 145 cm and a standard deviation of 20 cm. Find the number of boys, to the nearest whole number, with heights between:
125 \text{ cm} and 165 \text { cm}
105 cm and 185 cm
85 cm and 205 cm
145 cm and 165 cm
165 cm and 185 cm
The number of biscuits in a box is normally distributed with a mean of 30 and standard deviation of 3.
Approximately what percentage of scores lie between 2 standard deviations below and 1 standard deviation above the mean?
Hence, what percentage of the boxes have between 24 and 33 biscuits in them.
The weights of an adult harp seals are normally distributed with mean 144 kg and standard deviation of 6 kg.
Approximately 83.85\% of adult harp seals lie between 1 standard deviation below and how many standard deviation(s) above the mean?
Hence, approximately what percentage of adult males weigh between 138 kg and 162 kg?
The times for runners to complete a 100 m race is normally distributed with a mean of 14 seconds and a standard deviation of 1.9 seconds.
Approximately 97.35\% of people lie between 3 standard deviations below and how many standard deviation(s) above the mean?
Hence, approximately what percentage of people complete the race between 8.3 seconds and 17.8 seconds?
The number of biscuits packaged in biscuit boxes is normally distributed with mean 38 and standard deviation 5. If 4000 boxes of biscuits are produced, approximately how many boxes have more than 33 biscuits? Round your answer to the nearest integer.
The heights of players in a soccer club are 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 1.66 m? Round your answer to the nearest integer.
The times that professional divers can hold their breath are normally distributed with mean 106 seconds and standard deviation 8 seconds. If 700 professional divers are selected at random, approximately how many would be able to hold their breath for longer than 82 seconds? Round your answer to the nearest integer.
The heights of 400 netball players are normally distributed with a mean height of 149 cm and a standard deviation of 11. Find the number of players, to the nearest integer, that would be expected to have a height between:
138 \text{ cm} and 160 \text{ cm}
127 \text{ cm} and 171 \text{ cm}
116 \text{ cm} and 182 \text{ cm}
149 \text{ cm} and 160 \text{ cm}
160 \text{ cm} and 171 \text{ cm}
The marks in a class are normally distributed with a mean of 47 and a standard deviation of 6. Find the percentage of students that achieved:
A mark above the average
A mark above 41
A mark above 53
A mark below 35
A mark above 29
A mark between 29\text{ and } 47
A mark between 53 and 59
For a particular set of scores, it was found that the mean was 61 and the standard deviation was 11. If the scores are normally distributed, find the percentage of scores between:
50 and 72
39 and 83
28 and 94
61 and 72
72 and 83
The grades in a test are normally distributed. The mean mark is 60 with a standard deviation of 2. State which two scores the following percentage of results lie between, if the scores lie symmetrically about the mean:
The times that a class of students spent talking or texting on their phones on a particular weekend is normally distributed with a mean time of 173 minutes and standard deviation of 4 minutes. Approximately what percentage of students used their phones between 165 and 181 minutes on the weekend?
The height of sunflowers is normally distributed, with a mean height of 1.6 \text{ m} and a standard deviation of 5 \text{ cm}.
Approximately what percentage of sunflowers are between 1.5 m and 1.75 m tall?
Approximately what percentage of sunflowers are between 1.55 m and 1.75 m tall?
If there are 3000 sunflowers in the field, approximately how many are taller than 1.5 m? Round your answer to the nearest integer.
The exams scores of students are normally distributed with a mean score of 63 and a standard deviation of 8.
Approximately what percentage of students scored between 39 and 79?
There are 450 students in the class. If the passing score is 39, approximately how many students passed? Round your answer to the nearest whole number.
The operating times of phone batteries are normally distributed with a mean of 34 hours and a standard deviation of 4 hours.
What percentage of batteries last between 22 and 42 hours?
What percentage of batteries last between 30 hours and 42 hours?
Any battery that lasts less than 22 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.