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z Scores If your score on your next statistics test is converted to a z score, which of these z scores would you prefer: -2.00, -1.00, 0, 1.00, 2.00? Why?

Quadratic Mean The quadratic mean (or root mean square, or R.M.S.) is used in physical applications, such as power distribution systems. The quadratic mean of a set of values is obtained by squaring each value, adding those squares, dividing the sum by the number of values n, and then taking the square root of that result, as indicated below:

$\mathbf{Quadratic}\mathbf{mean}\mathbf{=}\sqrt{\frac{鈭�{\mathbf{x}}^{\mathbf{2}}}{\mathbf{n}}}$

Find the R.M.S. of these voltages measured from household current: 0, 60, 110, -110, -60, 0.

How does the result compare to the mean?

In Exercises 37鈥�40, refer to the frequency distribution in the given exercise and find the standard deviation by using the formula below, where x represents the class midpoint, f represents the class frequency, and n represents the total number of sample values. Also, compare the computed standard deviations to these standard deviations obtained by using Formula 3-4 with the original list of data values: (Exercise 37) 11.5 years; (Exercise 38) 8.9 years; (Exercise 39) 59.5; (Exercise 40) 65.4.

Standard deviation for frequency distribution

$s=\sqrt{\frac{n\left[鈭�\left(f脳x^2\right)\right]-{\left[鈭�\left(f脳x\right)\right]}^2}{n\left(n-1\right)}}$

The Empirical Rule Based on Data Set 1 鈥淏ody Data鈥� in Appendix B, blood platelet counts of women have a bell-shaped distribution with a mean of 255.1 and a standard deviation of 65.4. (All units are 1000 cells/L.) Using the empirical rule, what is the approximate percentage of women with platelet counts

a. within 2 standard deviations of the mean, or between 124.3 and 385.9?

b. between 189.7 and 320.5?

The Empirical Rule Based on Data Set 3 鈥淏ody Temperatures鈥� in Appendix B, body temperatures of healthy adults have a bell-shaped distribution with a mean of 98.20掳F and a standard deviation of 0.62掳F. Using the empirical rule, what is the approximate percentage of healthy adults with body temperatures

a. within 1 standard deviation of the mean, or between 97.58掳F and 98.82掳F?

b. between 96.34掳F and 100.06掳F?

Chebyshev鈥檚 Theorem Based on Data Set 1 鈥淏ody Data鈥� in Appendix B, blood platelet counts of women have a bell-shaped distribution with a mean of 255.1 and a standard deviation of 65.4. (All units are 1000 cells>L.) Using Chebyshev鈥檚 theorem, what do we know about the percentage of women with platelet counts that are within 3 standard deviations of the mean? What are the minimum and maximum platelet counts that are within 3 standard deviations of the mean?

Chebyshev鈥檚Theorem Based on Data Set 3 鈥淏ody Temperatures鈥� in Appendix B, body temperatures of healthy adults have a bell-shaped distribution with a mean of 98.20掳F and a standard deviation of 0.62掳F (using the data from 12 AM on day 2). Using Chebyshev鈥檚 theorem, what do we know about the percentage of healthy adults with body temperatures that are within 2 standard deviations of the mean? What are the minimum and maximum body temperatures that are within 2 standard deviations of the mean??

Why Divide by n 鈭� 1? Let a population consist of the values 9 cigarettes, 10 cigarettes, and 20 cigarettes smoked in a day (based on data from the California Health Interview Survey). Assume that samples of two values are randomly selected with replacement from this population. (That is, a selected value is replaced before the second selection is made.)

a. Find the variance${蟽}^2$ of the population {9 cigarettes, 10 cigarettes, 20 cigarettes}.

b. After listing the nine different possible samples of two values selected with replacement, find the sample variance $s^2$ (which includes division by n - 1) for each of them; then find the mean of the nine sample variances $s^2$.

c. For each of the nine different possible samples of two values selected with replacement, find the variance by treating each sample as if it is a population (using the formula for population variance, which includes division by n); then find the mean of those nine population variances.

d. Which approach results in values that are better estimates of${蟽}^2$ part (b) or part (c)? Why? When computing variances of samples, should you use division by n or n - 1?

e. The preceding parts show that $s^2$ is an unbiased estimator of ${蟽}^2$. Is s an unbiased estimator of $蟽$? Explain

Mean Absolute Deviation Use the same population of {9 cigarettes, 10 cigarettes, 20 cigarettes} from Exercise 45. Show that when samples of size 2 are randomly selected with replacement, the samples have mean absolute deviations that do not centre about the value of the mean absolute deviation of the population. What does this indicate about a sample mean absolute deviation being used as an estimator of the mean absolute deviation of a population?

Comparing Birth Weights The birth weights of a sample of males have a mean of 3272.8 g and a standard deviation of 660.2 g. The birth weights of a sample of females have a mean of 3037.1 g and a standard deviation of 706.3 g (based on Data Set 4 鈥淏irths鈥� in Appendix B). When considered among members of the same gender, which baby has the relatively larger birth weight: a male with a birth weight of 3400 g or a female with a birth weight of 3200 g? Why?