Question

a) What percentage of the area under the normal curve lies to the left of μ? % (b) What percentage of the area under the normal curve lies between μ − σ and μ + σ? % (c) What percentage of the area under the normal curve lies between μ − 3σ and μ + 3σ? %

Answer 1

a) 50%

b) 68%

c) 99%

Explanation:

for a standard normal curve ,

a) since the standard normal curve is symmetric and centred around μ , 50% of the curve lies at the left of μ and 50% lies to the right

b) according to the 68-95-99 rule, 68% of the standard normal curve lies from μ − σ and μ + σ

c) from the same rule , 99% of the standard normal curve lies from μ − 3σ and μ + 3σ

Answer 2

a) 50%

b) 68%

c) 99.7%

Explanation:

The Empirical Rule states that, for a normally distributed random variable:

68% of the measures are within 1 standard deviation of the mean.

95% of the measures are within 2 standard deviation of the mean.

99.7% of the measures are within 3 standard deviations of the mean.

The normal distribution is also symmetric, which means that 50% of the measures are below the mean and 50% are above.

In this problem, we have that:

Mean μ

Standard deviation σ

Area under the normal curve = percentage

a) What percentage of the area under the normal curve lies to the left of μ?

Normal distribution is symmetric, so the answer is 50%.

(b) What percentage of the area under the normal curve lies between μ − σ and μ + σ?

Within 1 standard deviation of the mean, so 68%.

(c) What percentage of the area under the normal curve lies between μ − 3σ and μ + 3σ?

Within 3 standard deviation of the mean, so 99.7%.