The Normal Distribution
The bell curve and its empirical rule, probabilities from Z-scores or technology, and 95% confidence bounds from a point estimate and standard error.
23 exercises
Exercise 1
Describe the shape of a Normal distribution.
Exercise 2
In a Normal distribution, what percentage of observations will lie to the left of the mean?
Exercise 3
If the diameters of trees in a forest follow a nearly Normal distribution has a mean of 35 inches and a standard deviation of 15 inches. What is the median width?
Exercise 4
Two Normal distributions have the same means but different standard deviations. Distribution A has a standard deviation of 10 inches, and distribution B has a standard deviation of 15 inches. Which curve has a wider spread along the horizontal axis? Why?
Exercise 5
Use the Empirical Rule to answer these questions.
About what percentage of the values from a Normal distribution fall within one standard deviation (left and right) of the mean?
Exercise 6
Use the Empirical Rule to answer these questions.
About what percentage of the values from a Normal distribution fall within two standard deviations (left and right) of the mean?
Exercise 7
Use the Empirical Rule to answer these questions.
About what percentage of the values from a Normal distribution fall within three standard deviations (left and right) of the mean?
Exercise 8
Use the Empirical Rule to answer these questions.
About what percentage of the values from a Normal distribution fall outside of three standard deviations (left and right) of the mean?
Exercise 9
Use the Empirical Rule to answer these questions.
About what percentage of the values from a Normal distribution fall between the first and second standard deviations from the mean (both sides)?
Exercise 10
Use the Empirical Rule to answer these questions.
About what percentage of the values from a Normal distribution fall between the first and third standard deviations (both sides)?
Exercise 11
Use the Empirical Rule to answer these questions.
About what percentage of values from a Normal distribution fall between the second and third standard deviations (on both sides)?
Exercise 12
Use the Empirical Rule to answer these questions.
About what percentage of values from a Normal distribution fall outside the first standard deviation from the mean (on both sides)?
Exercise 13
Use the Empirical Rule to answer these questions.
About what percent of values in a Normal distribution fall between the mean and one standard deviation below the mean?
Exercise 14
Use the Empirical Rule to answer these questions.
About what percent of values in a Normal distribution fall between the mean and three standard deviations above the mean?
Exercise 23
What does a z-score measure?
Exercise 25
For the distribution in problem 15, calculate the Z-score for a data value of 6.2 inches.
Exercise 26
For the distribution in problem 16, calculate the Z-score for a data value of 32 cm.
Exercise 27
For the distribution in problem 17, calculate the Z-score for a data value of $5.
Exercise 28
For the distribution in problem 18, calculate the Z-score for a data value of 19 miles.
Exercise 29
A random sample of 45 people who carry a purse found that they had an average of $2.35 in change in the bottom of their purse. The margin of error was $0.15. Calculate the 95% confidence interval and interpret the results.
Exercise 30
Scores on a certain quiz are normally distributed. In sample of 25 students the mean score was 14 points with a standard error estimate of 2 points. Calculate the margin of error and the 95% confidence intervals. Interpret Confidence interval.
Exercise 31
A researcher works on a study and found the sample mean to be 84.5 cm and the standard error of estimate to be 0.11 cm. What is the margin of error for the 95% confidence level? What is the 95% confidence interval for the true mean? Interpret the results.
Exercise 32
A researcher works on a study and found that the sample mean to be $35.4 and the standard error of estimate to be $0.75. What is the margin of error for the 95% confidence level? What is the 95% confidence interval for the true mean? Interpret the results.
Adapted from Math in Society © 2018–2024 Portland Community College Math Instructors. Licensed under CC BY-SA 4.0. Source: Math in Society. Imported at revision 68ee43c.