Question

The GMAC Insurance company reported that the mean score on the National Drivers Test was 70.5 with a standard deviation of 3.6 points. The test scores are approximately bell-shaped. Approximately 68% of all test scores were between two values A and B. What is the value of A? Write only a number as your answer. Round to one decimal place.

Answer 1

In statistics, the 68–95–99.7 rule, also known as the three-sigma rule or empirical rule, states that nearly all values lie within three standard deviations of the mean in a normal distribution.

About 68.27% of the values lie within one standard deviation of the mean. Similarly, about 95.45% of the values lie within two standard deviations of the mean. Nearly all (99.73%) of the values lie within three standard deviations of the mean.

In mathematical notation, these facts can be expressed as follows, where x is an observation from a normally distributed random variable, μ is the mean of the distribution, and σ is its standard deviation:

P( μ – σ ≤ x ≤ μ + σ ) ≈ 0.6827

P( μ –2σ ≤ x ≤ μ + 2σ ) ≈ 0.9545

P( μ –3σ ≤ x ≤ μ + 3σ ) ≈ 0.9973

So here A = μ – σ = 78.7 – 3.5 = 75.2 (answer