Exit Velocity Use the frequency distribution whose class width...

Question

Exit Velocity Use the frequency distribution whose class width is 4 obtained in Problem 25 in Section 2.2 to approximate the mean and standard deviation exit velocity. Compare these results to the actual mean and standard deviation exit velocity.

Accepted Answer

All right. Hello, everyone. So, this question says, a grouped frequency distribution for reaction times in milliseconds is listed as follows. Using class midpoints, bind the grouped sample mean and grouped sample standard deviation and select the correct pair. The pair in this case is X bar of the group and S of the group. And here we have 4 different answer choices labeled A through D. So first, let's begin with calculating our midpoints, which, if you recall, is equal to the lower limit of the class added to the upper limit and divided by 2. The midpoints Listed in order. By class are 712. 17 and 22. So now the total frequency. That is N is equal to the sum of all frequencies among all of the classes. So N is equal to the sum of 4. 7 6 and 3. Which gives you 2 From here we can use this to calculate the weighted sum. Which is the sum of all frequencies multiplied by their respective midpoints. I should move this off to the side here, here we go. So here, That would be 4 multiplied by 7. Added to 7 multiplied by 12. Added to 6 multiplied by 17. And added to 3 multiplied by 22. The weighted sum ultimately is 280. So from here we can calculate the group, the grouped mean. So X bar of the group. Is equal to. The weighted sum that we just found. Multiplied by the sum of all frequencies. So using what we already have, X bar of the group is equal to 280 divided by 20, which gives you 14. So in addition, we also have to find the contributions. Using the square deviation formula. So now I need to scroll down to open up some space. The contribution for each class is equal to the frequency of that particular class, multiplied by the difference between. The midpoint and the grouped mean. Squared So, for each class, the contributions are as follows. For the first, that's 4 multiplied by 7, subtracted by 14 squared, which gives you 196. For the second, that's 7 multiplied by 12 subtracted by 14 squared, which gives you 28. For the third, that 6 multiplied by 17 is subtracted by 14 squared, which gives you 54. And for the 4th class, that is 3 multiplied by 22, subtracted by 14 squared, which gives you 192. So the sum of all of your contributions is ultimately 470. And now this is what allows you to find the standard deviation of the group. But first, we can find the variants. Now the variance of the group. is equal to The sum of. All of the class contributions that you just calculated. Divided by The sum of all frequencies subtracted by one. So here, that's 470 divided by 20, subtracted by 1. Which is simply 470/19. So now, the standard deviation of the group is simply the square root of the variance. So that is the square root of 470/19. From here The pair That matches the values that we just calculated, happens to be option A, meaning that is our correct answer. And there you have it. So with that being said, thank you so very much for watching and I hope you found this helpful.

Exit Velocity Use the frequency distribution whose class width is 4 obtained in Problem 25 in Section 2.2 to approximate the mean and standard deviation exit velocity. Compare these results to the actual mean and standard deviation exit velocity.

Key Concepts

Frequency Distribution and Class Width

Approximating Mean and Standard Deviation from Grouped Data

Comparison of Approximate and Actual Statistics

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