Let x be a continuous random variable that is normally distributed with a mean of 65 and a standard deviation of 15. Find the probability that x assumes a value less than 44.
To find the probability that x assumes a value less than 44, we need to calculate the cumulative probability of x being less than 44 in a standard normal distribution.
Step 1: Standardize the value of 44.
The formula to standardize a value x is given by:
Z = (x - μ) / σ
Where Z is the standard score, x is the value we want to standardize, μ is the mean, and σ is the standard deviation.
In this case, x is 44, μ is 65, and σ is 15.
Z = (44 - 65) / 15
Z = -21 / 15
Z = -1.4
Step 2: Find the cumulative probability using the Z-table or statistical software.
The cumulative probability represents the probability of getting a value less than the standardized value of -1.4 in a standard normal distribution.
Using a Z-table or statistical software, we can find that the cumulative probability for Z = -1.4 is approximately 0.0808.
Step 3: Interpret the result.
The probability that x assumes a value less than 44 is approximately 0.0808, or 8.08%.
In this case, the probability can also be interpreted as the proportion of values in the normal distribution that are less than 44.
Note: If you are using a Z-table, you may need to interpolate between values to get a more accurate result.