Question

The income distribution in a county is a normal distribution with a mean income of $10,000. The top 2.5% of the wage earners earn $18,000 or more. Which sentence most closely summarizes this information? 68% of the wage earners earn less than $14,000 each. 16% of the wage earners earn less than $14,000 each. 55% of the wage earners earn less than $14,000 each. 84% of the wage earners earn less than $14,000 each. 95% of the wage earners earn less than $14,000 each.

Answer 1

Answer:

84% of the wage earners earn less than $14,000 each.

Step-by-step explanation:

The Empirical Rule (68-95-99.7%)-

According to this around 95% of the data will fall within two standard deviations of the mean.

As the bell curve is symmetrical, so the remaining 5% will be divided into 2 equal parts. So 2.5% will be above 2 standard deviation and 2.5% will be below 2 standard deviation.

As it is given that, the top 2.5% of the wage earners earn $18,000 or more, so 18,000 is 2 standard deviation away from the mean 10,000.

i.e $2\sigma=18000-10000=8000$

$\Rightarrow \sigma = 4000$

We know that,

$Z=\dfrac{X-\mu}{\sigma}$

where,

X = raw score = 14,000

μ = 10,000

σ = 4,000

Putting the values,

$Z=\dfrac{14000-10000}{4000}=\dfrac{4000}{4000}=1$

Now, calculating the value from the z score table,

$P(1)=0.8413=84.13\%$

As the probability at $z=1$ is the area below that, so 84% of the wage earners earn less than $14,000 each.