Final answer:
To find the probability P(74 < x < 81), convert the x-values to z-scores and use a standard normal distribution table or calculator to determine the area between the z-scores. This area represents the desired probability in the context of a normal distribution.
Step-by-step explanation:
The question asks us to find the probability P(74 < x < 81) for a normally distributed random variable x with a mean (μ) of 88 and a standard deviation (σ) of 5. To solve this, we can use a standard normal distribution table or a calculator with normal distribution functions, such as normalcdf. However, we need to convert our x-values to z-scores first.
To calculate the z-scores, we use the formula: z = (x - μ) / σ. For x = 74, z = (74 - 88) / 5 = -2.8. For x = 81, z = (81 - 88) / 5 = -1.4. Then we find the area between these two z-scores using the normal distribution, which represents the probability P(74 < x < 81).