3. Describing Data Numerically

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

Explain how the standard deviation measures dispersion. In your explanation, include a discussion of deviation about the mean.

Under which of the following conditions will a data value have a negative z-score when using the formula $z=\frac{x-\mu }{\sigma }$?

Finding Sample Statistics In Exercises 15 and 16, find the range, mean, variance, and standard deviation of the sample data set.

Pregnancy Durations The durations (in days) of pregnancies for a random sample of pregnant people

277 291 295 280 268 278 291

277 282 279 296 285 269 293

267 281 286 269 264 299 275

Which of the following is not a property of the $s$tandard deviation?

Answer the following based on the histograms shown in the next column.

a. Which measure of central tendency would you recommend reporting for the data whose histogram is shown in Figure I? Why? b. Which one has more dispersion? Explain.

Threaded Problem: Tornado The data set “Tornadoes_2017” located at www.pearsonhighered.com/sullivanstats contains a variety of variables that were measured for all tornadoes in the United States in 2017.

b. Determine the population mean and standard deviation length of all tornadoes in 2017.

z Scores. In Exercises 5–8, express all z scores with two decimal places.

New York City Commute Time New York City commute times (minutes) are listed in Data Set 31 “Commute Times” in Appendix B. The 1000 times have a mean of 42.6 minutes and a standard deviation of 26.2 minutes. Consider the commute time of 95.0 minutes.

b. How many standard deviations is that [the difference found in part (a)]?

Percentile Use the weights from Exercise 1 to find the percentile for 3647 mg.

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.

If the standard deviation for a set of data is $0$, what does this indicate about the data values?

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

Suppose the birth weights of newborns in a population are normally distributed with a mean of $3.5$ kg and a standard deviation of $0.5$ kg. What is the z-score of a newborn who weighs $4$ kg?

In Exercises 5–20, find the range, variance, and standard deviation for the given sample data. Include appropriate units (such as “minutes”) in your results. (The same data were used in Section 3-1, where we found measures of center. Here we find measures of variation.) Then answer the given questions.

Jaws 3 Listed below are the number of unprovoked shark attacks worldwide for the last several years. What extremely important characteristic of the data is not considered when finding the measures of variation?

70 54 68 82 79 83 76 73 98 81

Estimating the Sample Mean and Standard Deviation for Grouped Data In Exercises 41–44, make a frequency distribution for the data. Then use the table to estimate the sample mean and the sample standard deviation of the data set.

Weekly Study Hours The distribution of the number of hours that a random sample of college students study per week is shown in the pie chart. Use 32 as the midpoint for “30+ hours.”