A plate with an off-center hole is shown in Figure P7-36. Determine how close to
ID: 3008361 • Letter: A
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A plate with an off-center hole is shown in Figure P7-36. Determine how close to the top edge the hole can be placed before yielding of the A36 steel occurs (based on the maximum distortion energy theory). The applied tensile stress is 70,000 kPa, and the plate thickness is 6 mm. Now if the plate is made of 6061-T6 aluminum alloy with a yield strength of 255 MPa, does this change your answer? If the plate thickness is changed to 12 mm, how does this change the results? Use same total load as when the plate is 6 mm thick.Explanation / Answer
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The birth weights in pounds of five babies born one day in the same hospital are 9.2, 6.4, 10.5, 8.1, and 7.8. Obtain the sample mean and create a dot diagram. SOLUTION The mean birth weight for these data is x = 9.2 + 6.4 + 10.5 + 8.1 + 7.8 5 = 42.0 5 = 8.4 pounds The dot diagram of the data appears in Figure 9, where the sample mean (marked by ) is the balancing point or center of the picture. 6789 Pounds x 10 11 Figure 9 Dot diagram and the sample mean for the birth-weight data. Another measure of center is the middle value. The sample median of a set of n measurements x1,..., xn is the middle value when the measurements are arranged from smallest to largest. Roughly speaking, the median is the value that divides the data into two equal halves. In other words, 50% of the data lie below the median and 50% above it. If n is an odd number, there is a unique middle value and it is the median. If n is an even number, there are two middle values and the median is defined as their average. For instance, the ordered data 3, 5, 7, 8 have two middle values 5 and 7, so the median = ( 5 + 7 ) / 2 = 6. Example 8 Calculating the Sample Median Find the median of the birth-weight data given in Example 7. SOLUTION The measurements, ordered from smallest to largest, are 6.4 7.8 8.1 9.2 10.5 The middle value is 8.1, and the median is therefore 8.1 pounds. Example 9 Choosing between the Mean and Median Calculate the median of the survival times given in Example 5. Also calculate the mean and compare. DRAFT Johnson7e c02.tex V2 - 09/18/2013 3:51 P.M. Page 44 44 CHAPTER 2/ORGANIZATION AND DESCRIPTION OF DATA SOLUTION To find the median, first we order the data. The ordered values are 3 15 46 64 126 623 There are two middle values, so Median = 46 + 64 2 = 55 days The sample mean is x = 3 + 15 + 46 + 64 + 126 + 623 6 = 877 6 = 146.2 days Note that one large survival time greatly inflates the mean. Only 1 out of the 6 patients survived longer than x = 146.2 days. Here the median of 55 days appears to be a better indicator of the center than the mean. Example 9 demonstrates that the median is not affected by a few very small or very large observations, whereas the presence of such extremes can have a considerable effect on the mean. For extremely asymmetrical distributions, the median is likely to be a more sensible measure of center than the mean. That is why government reports on income distribution quote the median income as a summary, rather than the mean. A relatively small number of very highly paid persons can have a great effect on the mean salary. If the number of observations is quite large ( greater than, say, 25 or 30 ), it is sometimes useful to extend the notion of the median and divide the ordered data set into quarters. Just as the point for division into halves is called the median, the points for division into quarters are called quartiles. The points of division into more general fractions are called percentiles. The sample 100 p-th percentile is a value such that after the data are ordered from smallest to largest, at least 100p% of the observations are at or below this value and at least 100 ( 1 p ) % are at or above this value. If we take p = .5, the above conceptual description of the sample 100( .5 ) = 50th percentile specifies that at least half the observations are equal or smaller and at least half are equal or larger. If we take p = .25, the sample 100( .25 ) = 25th percentile has proportion one-fourth of the observations that are the same or smaller and proportion three-fourths that are the same or larger. We adopt the convention of taking an observed value for the sample percentile except when two adjacent values satisfy the definition, in which case their average is taken as the percentile. This coincides with the way the median is defined when the sample size is even. When all values in an interval satisfy the definition of a percentile, the particular convention used to locate a point in the interval does not appreciably alter the results in large data sets, except perhaps for DRAFT Johnson7e c02
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