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Answer :
Rounding to two decimal places, the answer is 0.32 or 32%. First, we need to calculate the mean and standard deviation of kinetic energy for the 25 throws:
Mean = (19.77 + 20.07 + 16.19 + 23.11 + 14.93 + 20.91 + 28.75 + 16.23 + 25.67 + 19.39 + 16.88 + 14.46 + 18.85 + 20.84 + 16.02 + 29.22 + 25.27 + 17.39 + 26.02 + 28.42 + 17.40 + 20.51 + 16.23 + 30.05 + 23.96) / 25
= 21.03 J
Standard deviation = sqrt[((19.77 - 21.03)^2 + (20.07 - 21.03)^2 + ... + (23.96 - 21.03)^2) / (25 - 1)]
= 4.40 J
To find the proportion of ball throws whose energy deviation from the mean is larger than the standard deviation, we need to first determine the cutoff values that define a deviation larger than one standard deviation from the mean. The lower cutoff is the mean minus one standard deviation, and the upper cutoff is the mean plus one standard deviation:
Lower cutoff = 21.03 - 4.40 = 16.63 J
Upper cutoff = 21.03 + 4.40 = 25.43 J
Next, we count how many of the 25 throws have a kinetic energy within this range:
Number of throws with energy deviation larger than one standard deviation = 8
Therefore, the point estimate of the proportion of all ball throws whose energy deviation from the mean is larger than the standard deviation is:
Proportion = Number of throws with energy deviation larger than one standard deviation / Total number of throws
= 8 / 25
= 0.32
Rounding to two decimal places, the answer is 0.32 or 32%.
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