2. A group of Brigham Young University-Idaho students (Matthew Herring. Nathan S

Question

2. A group of Brigham Young University-Idaho students (Matthew Herring. Nathan Spencer,Mark Walker, and Mark Steiner) collected data an the speed of vehidles traveing threugh a construct on zone on a state highway where the posted [oonstrction) speed Imit was 25 mph. The recorded speed for 14 randomiy selected vehides is given below 20 24 27 28 29 30 32 33 3 38 3 40 40 c. Determine the sample standard deviation of the speeds ld necesary. ruund to one more the data values). d. On the basis of the histogram erawn above, comment on the appropriateness of e the Empirical Rule to make any general comments about the drivers' speed e. Use the Empirical Rule to estimate the percentage of speeds that are between 26 and 38 mph the p f. Determine the actual percentage of drivers whose speeds are between 26 an d 38 mph. g. Use the Empirical Rule to estimate exceed the posted speed limit). te the percentage af speeds that are greater than 26 mph (so they af sp h. Determine the actual percentage of drivers whose speeds are 26 mph or higher

Explanation / Answer

c.
std. dev. = 6.1751

mean = 32.1429

e.
As 26 and 38 are located at 1-sigma level from the mean.

The empirical rule, also known as the three-sigma rule or the 68-95-99.7 rule, provides a quick estimate of the spread of data in a normal distribution given the mean and standard deviation. Specifically, the empirical rule states that for a normal distribution:

68% of the data will fall within one standard deviation of the mean.
95% of the data will fall within two standard deviations of the mean.
Almost all (99.7%) of the data will fall within three standard deviations of the mean.

Hence, 68% of spreads that are betwen 26 and 38 mph.

f)
P(26 < X < 38)
= P((26 - 32.1429)/6.1751 < z < (38 - 32.1429)/6.1751))
= P(-0.9948 < z < 0.9485)
= 0.6686

Hence, 66.86% of spreads that are betwen 26 and 38 mph.

g)
100 - 32/2 = 84% greater than 26 mph

h)
P(X > 26)
= P(z > (26 - 32.1429)/6.1751))
= P(z > -0.9948)
= 0.8401

i.e. 84.01%

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