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Professor smith took a sample of 214 students from her morning class and a sample of...

Professor smith took a sample of 214 students from her morning class and a sample of 115 students from her evening class. On Exam 1, all those sampled from the morning class got a 68, while those sampled from the evening class averaged a 68 with a variance of 158. If professor smith combines these two samples....

1) What is the combined Exam 1 Average?

2) What is the combined Exam 1 Median?

3) what is the shape of the combined Exam 1 grade distribution? (Skewed left, right or mound shape symmetrical)

4) What is the combined Exam 1 Standard Deviation?

(please explain how you got it! I need to learn this for the upcoming test)

math   Statistics-And-Probability   
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Answer #1

Let the averages of the sample from the morning class and the evening class be d T2 respectively. Let the variances of the sample from the morning class and the evening class be σ1 and σ2 respectively. Let the sizes of the sample from the morning class and the evening class be n1 and n2 respectively Given:-n1214 , nr 115, x2-68 , σ 58 Since the sample from the morning class all got the same marks 68, it is clear that 2 1-68 , ơi- 1) combined average = x = 1+n2x2-214 × 684 115 × 68 -68 ni+ n2 214 + 115

Total number of observations 214+115 329; which is a odd number So, after arranging the marks of 329 students in increasing order, combined median is the 165th observation [ (329+1%2-165] In the sample from the evening class, the average marks is 68. This implies that there are some students among the 115 students who got less marks than 68 and there are some students among the 115 students who got more marks than 68 Let the number of students from the sample from the evening class who got more marks than 68 be m [0<m115 Thus the number of students from the sample from the evening class who got marks less than or equal to 68 = 115-m < 115 Therefore in the combined sample, after arranging all the students with marks in increasing order the first (115-m) students have marks less than or equal to 68 the next 214 students all have marks equal to 68 . the last m students have marks more than 68 Therefore in the combined sample, total number of students who have marks less than or equal to 68 115-m+214 329-m Now; m<115 329-m>329-115329-m>114 Therefore in the combined sample, after arranging marks in increasing order;th student has marks less than or equal to 68. Also; 115-m<115115-m 114 mbined sam th student has marks equal to 68.3) There are at least 214 students with marks equal to 68, which implies that at most (329-214)-115 students have got marks other than 68. [115 214] Therefore, most of the students have got marks equal to 68 mode of the combined sampleE 68 Therefore the combined average,the combined median and the combined havethe same value,which is a feature of a symmetric distribution. . The shape of the combined distribution is mound shape symmetrical. 4) combined Uarzance n1 + n2 ni+ n2 214 x 0115 x 158 214(68 - 68)2 115(68 68)2 55.228 214 115 214 115 .:. combined standard deviation - V55.228-7.432

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