Question

Asked Oct 27, 2019

Use the Empirical Rule. The mean speed of a sample of vehicles along a stretch of highway is 69 miles per hour, with a standard deviation of 3 miles per hour. Estimate the percent of vehicles whose speeds are between 63 miles per hour and 75 miles per hour. (Assume the data set has a bell-shaped distribution.)

Approximately ___% of vehicles travel between 63 miles per hour and 75 miles per hour.

Step 1

**Solution:**

The empirical rule for any normal distribution is given below.

- 68% of the observations fall in the interval [µ –σ, µ +σ]: From the Empirical rule, 68% of the observations are lies between one standard deviation([µ ±σ). The minimum value corresponding with one standard deviations is µ–σ and the maximum value corresponding with one standard deviation µ+σ. This rule is called as 1-sigma rule.
- 95% of the observations fall in the interval[µ–2σ,µ +2σ]: The 95% of the observations are lies between two standard deviations ([µ ±2σ)The minimum value corresponding with two standard deviations is µ–2σ and the maximum value corresponding with two standard deviations µ+2σ. This rule is called as 2-sigma rule.
- 99.7% of the observations fall in the interval[µ–3σ,µ+3σ]: The 99.7% of the observations are lies between three standard deviations([µ ±3σ). The minimum value corresponding with three standard deviations is µ–3σ and the maximum value corresponding with three standard deviations is µ+3σ. This rule is called as 2-sigma rule.

Step 2

From the given information, the mean value is 69 miles and standard deviation value is 3 miles. The lower limit of interval is 63 miles per hour and upper limit of interval is 75 miles per hour.

The 1 sigma rule is,

[µ–σ,µ +σ]= [69–3,69 +3]=[66,72].

The 68% of vehicles travel between 66 miles per hour and 72 miles per hour.

The 2 sigma rule is,

[µ–2σ,µ +2σ]= [69–2x3,69 +2x3]=[63,75]

T...

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