2-5. Exercises

1. BASIC

Ex 1. State the Empirical Rule.

Ex 2. Describe the conditions under which the Empirical Rule may be applied.

Ex 3. State Chebyshev’s Theorem.

Ex 4. Describe the conditions under which Chebyshev’s Theorem may be applied.

Ex 5. A sample data set with a bell-shaped distribution has mean $\bar{x}=6$ and standard deviation $s = 2$ . Find the approximate proportion of observations in the data set that lie:

1. between 4 and 8;

2. between 2 and 10;

3. between 0 and 12.

Ex 6. A population data set with a bell-shaped distribution has mean $μ = 6$ and standard deviation $σ = 2$ . Find the approximate proportion of observations in the data set that lie:

1. between 4 and 8;

2. between 2 and 10;

3. between 0 and 12.

Ex 7. A population data set with a bell-shaped distribution has mean $μ = 2$ and standard deviation $σ = 1.1$ . Find the approximate proportion of observations in the data set that lie:

1. above 2;

2. above 3.1;

3. between 2 and 3.1.

Ex 8. A sample data set with a bell-shaped distribution has mean $\bar{x}=2$ and standard deviation $s = 1.1$ . Find the approximate proportion of observations in the data set that lie:

1. below −0.2;

2. below 3.1;

3. between −1.3 and 0.9.

Ex 9. A population data set with a bell-shaped distribution and size $N = 500$ has mean $μ = 2$ and standard deviation σ = 1.1. Find the approximate number of observations in the data set that lie:

1. above 2;

2. above 3.1;

3. between 2 and 3.1.

Ex 10. A sample data set with a bell-shaped distribution and size $n = 128$ has mean $\bar{x}=2$ and standard deviation $s = 1.1$ . Find the approximate number of observations in the data set that lie:

1. below −0.2;

2. below 3.1;

3. between −1.3 and 0.9.

Ex 11. A sample data set has mean $\bar{x}=6$ and standard deviation $s = 2$ . Find the minimum proportion of observations in the data set that must lie:

1. between 2 and 10;

2. between 0 and 12;

3. between 4 and 8.

Ex 12. A population data set has mean $μ = 2$ and standard deviation $σ = 1.1$ . Find the minimum proportion of observations in the data set that must lie:

1. between −0.2 and 4.2;

2. between −1.3 and 5.3.

Ex 13. A population data set of size N = 500 has mean $μ = 5.2$ and standard deviation $σ = 1.1$ . Find the minimum number of observations in the data set that must lie:

1. between 3 and 7.4;

2. between 1.9 and 8.5.

Ex 14. A sample data set of size n = 128 has mean $\bar{x}=2$ and standard deviation $s = 2$. Find the minimum number of observations in the data set that must lie:

1. between −2 and 6 (including −2 and 6);

2. between −4 and 8 (including −4 and 8).

Ex 15. A sample data set of size n = 30 has mean $\bar{x}=6$ and standard deviation $s = 2$ .

1. What is the maximum proportion of observations in the data set that can lie outside the interval (2,10)?

2. What can be said about the proportion of observations in the data set that are below 2?

3. What can be said about the proportion of observations in the data set that are above 10?

4. What can be said about the number of observations in the data set that are above 10?

Ex 16. A population data set has mean $μ = 2$ and standard deviation $σ = 1.1$ .

1. What is the maximum proportion of observations in the data set that can lie outside the interval $(−1.3, \space 5.3)$ ?

2. What can be said about the proportion of observations in the data set that are below −1.3?

3. What can be said about the proportion of observations in the data set that are above 5.3?

2. APPLICATIONS

Ex 17. Scores on a final exam taken by 1,200 students have a bell-shaped distribution with mean 72 and standard deviation 9.

1. What is the median score on the exam?

2. About how many students scored between 63 and 81?

3. About how many students scored between 72 and 90?

4. About how many students scored below 54?

Ex 18. Lengths of fish caught by a commercial fishing boat have a bell-shaped distribution with mean 23 inches and standard deviation 1.5 inches.

1. About what proportion of all fish caught are between 20 inches and 26 inches long?

2. About what proportion of all fish caught are between 20 inches and 23 inches long?

3. About how long is the longest fish caught (only a small fraction of a percent are longer)?

Ex 19. Hockey pucks used in professional hockey games must weigh between 5.5 and 6 ounces. If the weight of pucks manufactured by a particular process is bell-shaped, has mean 5.75 ounces and standard deviation 0.125 ounce, what proportion of the pucks will be usable in professional games?

Ex 20. Hockey pucks used in professional hockey games must weigh between 5.5 and 6 ounces. If the weight of pucks manufactured by a particular process is bell-shaped and has mean 5.75 ounces, how large can the standard deviation be if 99.7% of the pucks are to be usable in professional games?

Ex 21. Speeds of vehicles on a section of highway have a bell-shaped distribution with mean 60 mph and standard deviation 2.5 mph.

1. If the speed limit is 55 mph, about what proportion of vehicles are speeding?

2. What is the median speed for vehicles on this highway?

3. What is the percentile rank of the speed 65 mph?

4. What speed corresponds to the 16th percentile?

Ex 22. Suppose that, as in the previous exercise, speeds of vehicles on a section of highway have mean 60 mph and standard deviation 2.5 mph, but now the distribution of speeds is unknown.

1. If the speed limit is 55 mph, at least what proportion of vehicles must speeding?

2. What can be said about the proportion of vehicles going 65 mph or faster?

Ex 23. An instructor announces to the class that the scores on a recent exam had a bell-shaped distribution with mean 75 and standard deviation 5.

1. What is the median score?

2. Approximately what proportion of students in the class scored between 70 and 80?

3. Approximately what proportion of students in the class scored above 85?

4. What is the percentile rank of the score 85?

Ex 24. The GPAs of all currently registered students at a large university have a bell-shaped distribution with mean 2.7 and standard deviation 0.6. Students with a GPA below 1.5 are placed on academic probation. Approximately what percentage of currently registered students at the university are on academic probation?

Ex 25. Thirty-six students took an exam on which the average was 80 and the standard deviation was 6. A rumor says that five students had scores 61 or below. Can the rumor be true? Why or why not?

3. ADDITIONAL EXERCISES

Ex 26. For the sample data

x   26   27   28   29   30   31   32   
f    3    4   16   12    6    2    1

$Σx=1,256$ and $Σx^2=35,926$ .

1. Compute the mean and the standard deviation.

2. About how many of the measurements does the Empirical Rule predict will be in the interval (x−−−s,x−−+s)(x-−s,x-+s), the interval (x−−−2s,x−−+2s)(x-−2s,x-+2s), and the interval (x−−−3s,x−−+3s)(x-−3s,x-+3s)?

3. Compute the number of measurements that are actually in each of the intervals listed in part (a), and compare to the predicted numbers.

Ex 27. A sample of size n = 80 has mean 139 and standard deviation 13, but nothing else is known about it.

1. What can be said about the number of observations that lie in the interval (126,152)?

2. What can be said about the number of observations that lie in the interval (113,165)?

3. What can be said about the number of observations that exceed 165?

4. What can be said about the number of observations that either exceed 165 or are less than 113?

Ex 28. For the sample data

x    1    2    3    4   5
f   84   29    3    3   1

$Σx=168$ and $Σx^2=300$ .

1. Compute the sample mean and the sample standard deviation.

2. Considering the shape of the data set, do you expect the Empirical Rule to apply? Count the number of measurements within one standard deviation of the mean and compare it to the number predicted by the Empirical Rule.

3. What does Chebyshev’s Rule say about the number of measurements within one standard deviation of the mean?

4. Count the number of measurements within two standard deviations of the mean and compare it to the minimum number guaranteed by Chebyshev’s Theorem to lie in that interval.

Ex 29. For the sample data set

x  47 48 49 50 51
f   1  3 18  2  1

$Σx=1,224$ and $Σx^2=59,940$ .

1. Compute the sample mean and the sample standard deviation.

2. Considering the shape of the data set, do you expect the Empirical Rule to apply? Count the number of measurements within one standard deviation of the mean and compare it to the number predicted by the Empirical Rule.

3. What does Chebyshev’s Rule say about the number of measurements within one standard deviation of the mean?

4. Count the number of measurements within two standard deviations of the mean and compare it to the minimum number guaranteed by Chebyshev’s Theorem to lie in that interval.