25 athletes participate in a race and all of them finish it. The distribution of times taken to complete the race by athletes (in minutes) is shown in the table below. Calculate the mean of the finish times taken by the given population of athletes.
Transcribed Image Text:Finish time
(minutes)
0 to less than 1(0
10 to less than 20
20 to less than 30
30 to less than 40
40 to less than 50
Numberof
athletes
3
10
6
4
2
Definition Definition Measure of central tendency that is the average of a given data set. The mean value is evaluated as the quotient of the sum of all observations by the sample size. The mean, in contrast to a median, is affected by extreme values. Very large or very small values can distract the mean from the center of the data. Arithmetic mean: The most common type of mean is the arithmetic mean. It is evaluated using the formula: μ = 1 N ∑ i = 1 N x i Other types of means are the geometric mean, logarithmic mean, and harmonic mean. Geometric mean: The nth root of the product of n observations from a data set is defined as the geometric mean of the set: G = x 1 x 2 ... x n n Logarithmic mean: The difference of the natural logarithms of the two numbers, divided by the difference between the numbers is the logarithmic mean of the two numbers. The logarithmic mean is used particularly in heat transfer and mass transfer. ln x 2 − ln x 1 x 2 − x 1 Harmonic mean: The inverse of the arithmetic mean of the inverses of all the numbers in a data set is the harmonic mean of the data. 1 1 x 1 + 1 x 2 + ...
Expert Solution
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Step 1
This is a problem on finding the mean of continuous grouped data. We first find the midpoint of each interval. For example, the first interval or class is 0 - 10. The mid-point of this class will be (0 + 10)/2 = 5.
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Step 2
We now calculate the product of mid-point and frequency for each class. Here, the number of athletes in each class is the frequency of that class. For example, for the first class 0 – 10, its frequency is 3.
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Follow-up Questions
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Follow-up Question
Suppose another researcher takes a random sample of 1000 subjects from the same population (US physicians who received payments from any of 8 major pharmaceutical companies in 2011). You have not yet seen these data. Likely, how will the sample standard deviation of these 1000 values (s1000) compare to the sample standard deviation of individual physician payments in the sample of 400 (s400=17 )
Read through expert solutions to related follow-up questions below.
Follow-up Question
Suppose another researcher takes a random sample of 1000 subjects from the same population (US physicians who received payments from any of 8 major pharmaceutical companies in 2011). You have not yet seen these data. Likely, how will the sample standard deviation of these 1000 values (s1000) compare to the sample standard deviation of individual physician payments in the sample of 400 (s400=17 )
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