Exercises 10.4 Exercises
Exercise Group.
For the exercises below, describe the rarity of an event using the following scale:
0% - < 1% probability = very rare
1% - < 5% = rare
5% - < 34% = uncommon
34% - < 50% = common
50% - 100% = likely
1.
Scores on a certain standardized test have a mean of 500, and a standard deviation of 100. How common is a score between 600 and 700?
2.
Considering a full-grown show-quality male Siberian Husky has a mean weight of 52.5 lbs, with SD of 7.5 lbs, how common are male huskies in the 37.5 - 45 lbs range?
3.
In a population with \(\mu=125\text{,}\) and \(\sigma=25\text{,}\) how common are values in the \(125 - 150\) range?
4.
In a population with \(\mu =0.0025\) and \(\sigma=0.0005\text{,}\) how common are values between \(0.0025\) and \(0.0030\text{?}\)
5.
A 12 oz can of soda has a mean volume of 12 oz, with a standard deviation of .25 oz. How common are cans with between 11 and 11.5 oz of soda?
6.
If \(\mu = 0.0025\) and \(\sigma = 0.0005\text{,}\) how common are values between 0.0045 and 0.005?
7.
The American Robin Redbreast has a mean weight of 77 g, with a standard deviation of 6 g. How common are Robins in the 59 g-71 g range?
8.
Assuming population has a mean of 1130 and a standard deviaton of 5, how common are values between 1125 and 1135?
9.
A population has \(\mu=\frac{3}{5}\) and \(\sigma=\frac{1}{10}\text{,}\) how common are values between \(\frac{2}{5}\) and \(1\text{?}\)
10.
Given mean 63 and standard deviation of 168, find the approximate percentage of the distribution that lies between -105 and 567.
11.
Given standard deviation of 74 and mean of 124, approximately what percentage of the values are greater than 198?
12.
Given \(\sigma = 39\) and \(\mu = 101\text{,}\) approximately what percentage of the values are less than \(23\text{?}\)
13.
Given a mean of 92 and standard deviation of 189, find the approximate percentage of the distribution that lies between -286 and 470.
14.
Approximately what percent of a normal distribution lies between \(\mu + 1\sigma\) and \(\mu + 2\sigma\text{?}\)
15.
Given standard deviation of 113 and mean 81, approximately what percentage of the values are less than -145?
16.
Given mean 23 and standard deviation 157, find the approximate percentage of the distribution that lies between 23 and 337.
17.
Given \(\sigma = 3\) and \(\mu = 84\text{,}\) approximately what percentage of the values are less than \(90\text{?}\)
18.
Approximately what percent of a normal distribution is between \(\mu\) and \(\mu + 1\sigma\text{?}\)
19.
Given mean 118 and standard deviation 145, find the approximate percentage of the distribution that lies between -27 and 118.
20.
Given standard deviation of 81 and mean 67, approximately what percentage of values are greater than 310?
21.
Approximately what percent of a normal distribution is less than 2 standard deviations from the mean?
22.
Given \(\mu + 1\sigma = 247\) and \(\mu + 2\sigma = 428\text{,}\) find the approximate percentage of the distribution that lies between \(66\) and \(428\text{.}\)
23.
Given \(\mu - 1\sigma = -131\) and \(\mu + 1\sigma = 233\text{,}\) approximately what percentage of the values are greater than \(-495\text{?}\)
24.
Suppose people’s heights are normally distributed with a mean of 66 inches and a standard deviation of 3 inches.
What percent of people would you expect to be taller than 66 inches?
What percent of people would you expect to be taller than 69 inches?
What percent of people would you expect to be taller than 72 inches?
25.
300 students take a final exam. The scores are normally distributed with a mean of 75 and a standard deviation of 9. Use the 68-95-99.7 rule to determine:
How many students scored more than 75 points?
How many students scored more than 84 points?
How many students scored between 66 and 84 points?
2.5% of the students received a score above what value?
26.
The life of batteries are normally distributed with a mean of 215 hours and a standard deviation of 21 hours. Use the 68-95-99.7 rule to determine:
If you randomly select a battery, what is the probability that is will last more than 257 hours?
If a shipment of 500 batteries, how many would you expect to last less than 194 hours?
27.
Birth weight is normally distributed with a mean of 6.5 pounds and a standard deviation of 1.5 pounds. Last year, 445 babies were born at a hospital. Use the 68-95-99.7 rule to determine:
How many babies would you expect to weigh more than 8 pounds?
How many babies would you expect to weigh less than 6.5 pounds?
How many babies would you expect to weigh between 6.5 and 8 pounds?
How many babies would you expect to weigh more than 9.5 pounds?
71 babies
223 babies
151 babies
11 babies
28.
The number of candies in machine-filled boxes are normally distributed with a mean of 30 candies and a standard deviation of 1 candy. Use the 68-95-99.7 rule to determine:
What is the probability that a box of candies will contain more than 32 candies?
What percent of boxes would you expect to contain more than 27 candies?
What percent of boxes would you expect to contain between 32 and 33 candies?
In a shipment of 2,000 boxes, how many would you expect to have less than 28 candies?
29.
Given a distribution with a mean of 70 and standard deviation of 62, find a value with a z-score of -1.82.
30.
What does a z-score of 3.4 mean?
31.
Given a distribution with a mean of 60 and standard deviation of 98, find the z-score of 120.76.
32.
Given a distribution with a mean of 60 and standard deviation of 21, find a value with a z-score of 2.19.
33.
Find the z-score of 187.37, given a distribution with a mean of 185 and standard deviation of 1.
34.
Find the z-score of 125.18, given a distribution with a mean of 101 and standard deviation of 62.
35.
What does a z-score of -3.8 mean?
36.
Given a distribution with a mean of 117 and standard deviation of 42, find a value with a z-score of -0.94.
37.
Given a distribution with a mean of 126 and standard deviation of 100, find a value with a z-score of -0.75.
38.
Find the z-score of 264.16, given \(\mu = 188\) and \(\sigma = 64\text{.}\)
39.
Find a value with a z-score of -0.2, given \(\mu = 145\) and \(\sigma = 56\text{.}\)
40.
Find a value with a z-score of 1.2, given \(\mu = 48\) and \(\sigma = 5\text{.}\)
Exercise Group.
41.
What is the probability of a z-score less than +2.02?42.
What is the probability of a z-score less than -1.97?43.
What is the probability of a z-score greater than +2.02?44.
What is the probability of a z-score greater than -1.97?45.
What is the probability of a z-score less than -0.02?46.
What is the probability of a z-score less than +0.09?47.
What is the probability of the random occurrence of a value greater than 56 from a normally distributed population with mean 62 and standard deviation 4.5?48.
What is the probability of a value of 329 or greater, assuming a normally distributed set with mean 290 and standard deviation 32?49.
What is the probability of getting a value below 1.2 from the random output of a normally distributed set with \(\mu = 2.6\) and \(\sigma = .9\text{?}\)Exercise Group.
50.
What is the probability of a z-score between +1.99 and +2.02?51.
What is the probability of a z-score between -1.99 and +2.02?52.
What is the probability of a z-score between -1.20 and -1.97?53.
What is the probability of a z-score between +2.33 and -0.97?54.
What is P(1.42 < Z < 2.01)?55.
What is P(1.77 < Z < 2.22)?56.
What is P(-2.33 < Z < -1.19)?57.
What is the probability of the random occurrence of a value between 56 and 61 from a normally distributed population with mean 62 and standard deviation 4.5?58.
What is the probability of a value between 301 and 329, assuming a normally distributed set with mean 290 and standard deviation 32?59.
40% of the data in a normal disribution is above what z-score?60.
30% of the data in a normal disribution is above what z-score?61.
The annual rainfall in a city is normally distributed with a mean of 28.4 inches and a standard deviation of 1.7 inches. What is the probability that a year will have less than 24 inches of rain?62.
The diameter of a pipe is normally distributed with a mean of 0.5 inches and a standard deviation of 0.04 inches. What is the probability that the diameter of a randomly selected pipe will exceed 0.6 inches?63.
The lifetime of a light bulb is normally distributed with a mean of 95 hours and a standard deviation of 12 hours.
What percentage of light bulbs will last more than 100 hours?
What percentage of light bulbs will last less than 75 hours?
33.72%
4.75%
64.
Grades on a test are normally distributed with a mean of 50 pts and a standard deviation of 8 pts. In a class of 150 students,
How many students scored more than 60 pts?
How many students scored between 40 and 60 pts
65.
A pizza place wants to give coupons those with the top 10% of wait times for their deliveries. Assuming the wait times are normally distributed with a mean delivery time of 30 minutes and a standard deviation of 8 minutes, what delivery times will earn coupons?
66.
Race times are normally distributed with a mean of 23 minutes and a standard deviation of 4 minutes. If the top 20% of finishers will move on to the State Championship, what is the cutoff time for the State Championship?
67.
A pizza place has a mean delivery time of 30 minutes with a standard deviation of 8 minutes. Assuming the times are normally distributed, what is the probability that your pizza will be delivered in less than 20 minutes?