3. Describing Data Numerically

The Empirical Rule SAT Math scores have a bell-shaped distribution with a mean of 515 and a standard deviation of 114.

Source: College Board

c. What percentage of SAT scores is greater than 743?

Exercises 1–10 are based on the following sample data consisting of costs of dinner (dollars) and the amounts of tips (dollars) left by diners. The data were collected by students of the author.

Predictions The sample data result in a linear correlation coefficient of r = 0.846 and the regression equation y^ = -0.00777 + 0.145x. What is the best predicted amount of tip, given that the cost of dinner was \$84.62? How was the predicted value found?

Given the data set $4,8,12,24$, what is the standard deviation of the data (rounded to two decimal places)?

In statistics, which symbol is commonly used to represent the standard error of the sample mean $m$?

As the mean $\mu$ increases in a normal distribution, what happens to the graph of the normal curve?

Which formula should you use to calculate the variance and which formula should you use to calculate the standard deviation of a sample of $n$ observations $x_1$, $x_2$, ..., $x_n$?

Which of the following is true about $\mathrm{variance}$?

What is meant by the phrase degrees of freedom as it pertains to the computation of the sample standard deviation?

Which of the following best describes the relationship between $\mathrm{variance}$ and $\mathrm{standard\; deviation}$ for a data set?

Which three quantities are required to calculate the variance of a data set?

Which of the following is the correct formula for the population variance?

Identifying Significant Values with the Range Rule of Thumb. In Exercises 33–36, use the range rule of thumb to identify the limits separating values that are significantly low or significantly high.

U.S. Presidents Based on Data Set 22 “Presidents” in Appendix B, at the time of their first inauguration, presidents have a mean age of 55.2 years and a standard deviation of 6.9 years. Is the minimum required 35-year age for a president significantly low?

Which Professor? Suppose Professor Alpha and Professor Omega each teach Introductory Biology. You need to decide which professor to take the class from and have just completed your Introductory Statistics course. Records obtained from past students indicate that students in Professor Alpha’s class have a mean score of 80% with a standard deviation of 5%, while past students in Professor Omega’s class have a mean score of 80% with a standard deviation of 10%. Decide which instructor to take for Introductory Biology using a statistical argument.

Savings Recently, a random sample of 25–34 year olds was asked, “How much do you currently have in savings, not including retirement savings?” The following data represent the responses to the survey. Approximate the mean and standard deviation amount of savings.

Constructing Data Sets In Exercises 25–28, construct a data set that has the given statistics.

N = 6

μ = 5

σ ≈ 2