3. Describing Data Numerically

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.

Comparing Standard Deviations The standard deviation of batting averages of all teams in the American League is 0.008. The standard deviation of all players in the American League is 0.02154. Why is there less variability in team batting averages?

Mean Absolute Deviation Another measure of variation is the mean absolute deviation. It is computed using the formula:

MAD = (Σ |xi − x̄|) / n

Compute the mean absolute deviation of the data in Problem 11 and compare the results with the sample standard deviation.

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.

Triathlon Roberto finishes a triathlon (750-meter swim, 5-kilometer run, and 20-kilometer bicycle) in 63.2 minutes. Among all men in the race, the mean finishing time was 69.4 minutes with a standard deviation of 8.9 minutes. Zandra finishes the same triathlon in 79.3 minutes. Among all women in the race, the mean finishing time was 84.7 minutes with a standard deviation of 7.4 minutes. Who did better in relation to their gender?

Develop a sample of size n = 8 such that x̄ = 15 and s = 0.

Household Winter Temperature Often, frequency distributions are reported using unequal class widths because the frequencies of some groups would otherwise be small or very large. Consider the following data, which represent the daytime household temperature the thermostat is set to when someone is home for a random sample of 750 households. Determine the class midpoint, if necessary, for each class and approximate the mean and standard deviation temperature.

Birth Weights Babies born after a gestation period of 32–35 weeks have a mean weight of 2600 grams and a standard deviation of 660 grams. Babies born after a gestation period of 40 weeks have a mean weight of 3500 grams and a standard deviation of 470 grams. Suppose a 34-week gestation period baby weighs 3000 grams and a 40-week gestation period baby weighs 3900 grams. What is the z-score for the 34-week gestation period baby? What is the z-score for the 40-week gestation period baby? Which baby weighs less relative to the gestation period?

ERA Champions In 2018, Jacob deGrom of the New York Mets had the lowest earned-run average (ERA is the mean number of runs yielded per nine innings pitched) of any starting pitcher in the National League, with an ERA of 1.70. Also in 2018, Blake Snell of the Tampa Bay Rays had the lowest ERA of any starting pitcher in the American League with an ERA of 1.89. In the National League, the mean ERA in 2018 was 3.611 and the standard deviation was 0.772. In the American League, the mean ERA in 2018 was 3.744 and the standard deviation was 0.893. Which player had the better year relative to his peers? Why?

Quality Control A manufacturer of bolts has a qualitycontrol policy that requires it to destroy any bolts that are more than 2 standard deviations from the mean. The quality-control engineer knows that the bolts coming off the assembly line have a mean length of 8 cm with a standard deviation of 0.05 cm. For what lengths will a bolt be destroyed? 167

Travel Time to Work The frequency distribution listed in the table represents the travel time to work (in minutes) for a random sample of 895 U.S. adults.

b. Approximate the standard deviation travel time to work for U.S. adults.

For each of the following data sets, decide which has the higher standard deviation (set 1 or set 2), if any, without doing any computation. Explain the rationale behind your choice. Then, verify your choice by computing the standard deviation by hand.

pH in Water The acidity or alkalinity of a solution is measured using pH. A pH less than 7 is acidic, while a pH greater than 7 is alkaline. The following data represent the pH in samples of bottled water and tap water. a. Which type of water has more dispersion in pH using the range as the measure of dispersion?

pH in Water The acidity or alkalinity of a solution is measured using pH. A pH less than 7 is acidic, while a pH greater than 7 is alkaline. The following data represent the pH in samples of bottled water and tap water. b. Which type of water has more dispersion in pH using the standard deviation as the measure of dispersion?

Soybean Yield The data in the next column represent the number of pods on a sample of soybean plants for two different plot types. Which plot type do you think is superior? Why?