Prove the following proposition: For all integers a and b, if 3 divides (a² + b²), then 3 divides a and 3 divides b.

Can people tell the difference between a female nose and a male nose? This research question was examined in an article. Eight Caucasian males and eight Caucasian females posed for nose photos. The article states that none of the volunteers wore nose studs or had prominent nasal hair. Each person placed a black Lycra tube over his or her head in such a way that only the nose protruded through a hole in the material. Photos were then taken from three different angles: front view, three-quarter view, and profile. These photos were shown to a sample of undergraduate students. Each student in the sample was shown one of the nose photos and asked whether it was a photo of a male or a female. The response was then classified as either correct or incorrect. The following table was constructed using summary values reported in the article. Sex ID View Front Profile Three-Quarter Correct 29 26 23 Incorrect 11 14 17   Is there evidence that the proportion of correct sex…

Write out solutions to the following questions. Show all calculations using the formulas. Include all diagrams appropriately labeled. Use proper notation throughout the solution. Think of the assignment as if you were demonstrating to someone how to do these problems. The quality of the work and the neatness and organization of the work will count in the grade.   A random sample of 25 frozen dinners of a certain type was selected from production during a particular period and the calorie content of each one was determined.  The resulting observations are: Calories from Frozen Dinners 255 244 239 242 265 245 259 248 225 226 251 233 266 227 246 240 257 222 237 270 230 260 268 231 238 The stated calorie content is 240 calories.  At the 5% significance level, do the data suggest that the mean calorie content of the frozen dinners differs from 240 calories? State the null and alternative hypothesis Check the conditions.  To determine if the data is approximately normal, using…

Find R in the triangle below, if X = 6 ft, Y = 6.4 ft, α 219 R Y