Question

when hudson goes bowling, his scores are normally distributed with a mean of 185 and a standard deviation of 14. what is the probability that the next game hudson bowls, his sco be less than 160, to the nearest thousandth? statistics calculator

Explanation:

Step1: Calculate the z-score

The formula for the z - score is $z=\frac{x - \mu}{\sigma}$, where $x$ is the value from the dataset, $\mu$ is the mean, and $\sigma$ is the standard deviation.
Given that $\mu = 185$, $\sigma=14$, and $x = 160$.
Substitute these values into the formula: $z=\frac{160 - 185}{14}=\frac{- 25}{14}\approx - 1.7857$

Step2: Find the probability using the z - score

We need to find $P(Z < - 1.7857)$ for a standard normal distribution (where $Z$ follows a normal distribution with mean 0 and standard deviation 1).
Using a standard normal table or a calculator with a normal - distribution function, we can find the probability corresponding to $z=-1.7857$.
Using a calculator (for example, the normalcdf function on a TI - 84 plus, where we can enter normalcdf(-10000, - 1.7857, 0, 1) since we want the area to the left of $z=-1.7857$ in a standard normal distribution), we get the probability.
Using a standard normal table or a more precise calculation, we find that $P(Z < - 1.7857)\approx0.037$

Answer:

0.037