The profit of a small and medium enterprise (SME) has a mean of #£46,000# and standard deviation of #£19,000#. What is the probability that profit of a SME will be between #£40,000# and #£50,000#?

Question

Elena Atkins · Accepted Answer

The probability that a specific SME will have a profit between #£40,000# and #£50,000# is #0.2073# or #20.73%# As mean is #mu=£46,000# and standard deviation #sigma=£19,000#, the #z#-score for #£40,000# is #(40000-46000)/19000=-0.3158# and #z#-score for #£50,000# is #(50000-46000)/19000=0.2105# From tables the probability between #z=-0.3158# and #z=0# is #0.1239# and probability between #z=0# and #z=0.2105# is #0.0834# Hence, the probability that a specific SME will have a profit between #£40,000# and #£50,000# is #0.1239+0.0834=0.2073# or #20.73%#

Audrey Andrews · Answer

undefined To find the probability that the profit of an SME will be between £40,000 and £50,000, we can use the standard normal distribution since we know the mean and standard deviation.

First, we need to standardize the values £40,000 and £50,000 using the z-score formula:

$$ z = \frac{x - \mu}{\sigma} $$

Where:
- $ x $ is the value (£40,000 or £50,000)
- $ \mu $ is the mean (£46,000)
- $ \sigma $ is the standard deviation (£19,000)

For £40,000:
$$ z_1 = \frac{40,000 - 46,000}{19,000} = -0.3158 $$

For £50,000:
$$ z_2 = \frac{50,000 - 46,000}{19,000} = 0.2105 $$

Now, we need to find the cumulative probability associated with these z-scores using a standard normal distribution table or a calculator.

Let's denote $ P(z_1) $ as the cumulative probability associated with $ z_1 $ and $ P(z_2) $ as the cumulative probability associated with $ z_2 $.

Then, the probability that the profit of an SME will be between £40,000 and £50,000 is:

$$ P(40,000 < X < 50,000) = P(z_2) - P(z_1) $$

Substituting the calculated values of $ z_1 $ and $ z_2 $, we can find $ P(40,000 < X < 50,000) $.