Based off these numbers, answer a-c a) Using your sample mean, sample standard d

ID: 3374442 • Letter: B

Question

Based off these numbers, answer a-c

a) Using your sample mean, sample standard deviation, and histogram to help, what is the approximate theoretical distribution of the data?

X ~_(_,_)

Use your approximate theoretical distribution to answer the following:

The 15th percentile is:___

The 85th percentile is:___

Explain the meaning of the 85th percentile of this data:________

The median is:___

The empirical probability that a randomly chosen lap time is more than 130 seconds is _______

c) What is more likely and why:

For a lap to take 110 seconds or to take 135 seconds?

134 129 131 134 126 129 128 132 130 125 131 130 127 130 132 129 132 131 126 133 132 130 129 127 129 131 128 127 139 132 130 133 129 129 128 131

a) Mean = 130.03 , Standard deviation = 2.7

Therefore, theoretical distribution = X ~ (130.03, 2.7)

b) The 15th percentile is that value that is higher than 15% of the data or where P(X < x) = 0.15

From the Z-table : Z-score = -1.05 is the 15th percentile in a standard normal distribution.

Z = -1.05 --->

Z = X - Mean / SD

-1.05 = X - 130.03 / 2.7

X = -127.19

Therefore, 15th percentile = -127.19

The 85th percentile is that value that is higher than 85% of the data or where P(X < x) = 0.85

From the Z-table : Z-score = 1.05 is the 85th percentile in a standard normal distribution.

Z = 1.05 --->

Z = X - Mean / SD

1.05 = X - 130.03 / 2.7

X = 132.865

Therefore, 85th percentile = 132.865

In a normal distribution, the mean = median and so median = 130.03

P(X > 130) ---->

Z = 130 - 130.03 / 2.7

Z = -0.01

P(X > 130) = P(Z > -0.01) = 1 - P(Z < -0.01) = 1 - 0.4960 = 0.5040

135 seconds is closer to the mean = 130.03, and so, a lap to take 135 seconds is more likely.

Cheers!

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