Answer: The Empirical Rule The following data represent

Answer :

Step 1

First we need to find mean.

 = 

 = 

 = 1041.13

Here we have to find the standard deviation.

The formula of the standard deviation is

= 

Here n=44

Then,

= 

= 36

Then the sample variance is

s=

s =

s = 6

Hence the sample standard deviation length of eruption is s = 6.

b)

The histogram is given below.

Hence the histogram is approximately symmetric, so the Empirical Rule is applicable.

c)

Lower bound: 92 = 104 – 2(6) which is 2 standard deviations below the mean ( -2s).

Upper bound: 116 = 104 + 2(6) which is 2 standard deviations above the mean ( +2s). According to the empirical rule, 95% of the data fall within ().

95% of the eruptions last between 92 and 116 seconds.

95% of the eruptions should last between 92 and 116 seconds.

d)

Part (c) is the theoretical percentage obtained from Empirical rule. Part (d) asked for the actual percentage of the given raw data that fall with () .

We will first sort the data in a ascending order then count how many data actually fall between 92 and 116 seconds inclusive.

93% of the observed eruptions actually lasted between 92 and 116 seconds.

e)

This part does not require the use of StatCrunch. The calculation is done by hand based on the Empirical Rule. x =104 and s =6 98 = 104 - 6 which is one standard deviation below the mean

( -1s).

According to the Empirical rule, 68% of the data fall within ().

Thus 32% (100% - 68%) fall outside ().

Due to the symmetrical shape of a bell-shape curve, 16% of the data fall below () 1 and 16% of the data fall above ()  .

16% of the eruptions should last less than 98 seconds.

f)

We just have to look at the sorted data and count the number of eruptions that last less than 98 seconds. There are 5 out of 44 eruptions had the length of eruption that last less than 98 seconds. Thus the actual percentage is (5/44) x 100 11.4%.

11% of the observed eruptions actually lasted fewer than 98 seconds.

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