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Chapter 3: Problem 3
Using an example, show how outliers can affect the value of the mean.
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Suppose that on a certain section of I-95 with a posted speed limit of \(65 \mathrm{mph}\), the speeds of all vehicles have a bell-shaped distribution with a mean of \(72 \mathrm{mph}\) and a standard deviation of \(3 \mathrm{mph}\). a. Using the empirical rule, find the percentage of vehicles with the following speeds on this section of I-95. i. 63 to \(81 \mathrm{mph}\) ii. 69 to \(75 \mathrm{mph}\) *b. Using the empirical rule, find the interval that contains the speeds of \(95 \%\) of vehicles traveling on this section of \(\mathrm{I}-95\).
Prepare a box-and-whisker plot for the following data: \(\begin{array}{lrrrrrrrrr}11 & 8 & 26 & 31 & 62 & 19 & 7 & 3 & 14 & 75 \\ 33 & 30 & 42 & 15 & 18 & 23 & 29 & 13 & 16 & 6\end{array}\) Does this data set contain any outliers?
Assume that the annual earnings of all employees with CPA certification and 6 years of experience and working for large firms have a bell-shaped distribution with a mean of \(\$ 134,000\) and a standard deviation of \(\$ 12,000\). a. Using the empirical rule, find the percentage of all such employees whose annual earnings are hetween i. \(\$ 98,000\) and \(\$ 170,000\) ii. \(\$ 110,000\) and \(\$ 158,000\) "b. Using the empirical rule, find the interval that contains the annual earnings of \(68 \%\) of all such employees.
The following data give the hourly wage rates of eight employees of a company. \(\begin{array}{llllllll}\$ 22 & 22 & 22 & 22 & 22 & 22 & 22 & 22\end{array}\) Calculate the standard deviation. Is its value zero? If yes, why?
One property of the mean is that if we know the means and sample sizes of two (or more) data sets, we can calculate the combined mean of both (or all) data sets. The combined mean for two data sets is calculated by using the formula $$ \text { Combined mean }=\bar{x}=\frac{n_{1} \bar{x}_{1}+n_{2} \bar{x}_{2}}{n_{1}+n_{2}} $$ where \(n_{1}\) and \(n_{2}\) are the sample sizes of the two data sets and \(\bar{x}_{1}\) and \(\bar{x}_{2}\) are the means of the two data sets, respectively. Suppose a sample of 10 statistics books gave a mean price of \(\$ 140\) and a sample of 8 mathematics books gave a mean price of \(\$ 160\). Find the combined mean. (Hint: For this example: \(\left.n_{1}=10, n_{2}=8, \bar{x}_{1}=\$ 140, \bar{x}_{2}=\$ 160 .\right)\)
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