Question 1191348
Here's how to solve this problem:

1. **Find the standard error of the mean:**

The standard error of the mean (SEM) is the standard deviation of the sample means.  It's calculated as:

SEM = σ / √n

Where:
* σ is the population standard deviation (93 hours)
* n is the sample size (23 bulbs)

SEM = 93 / √23 ≈ 19.38

2. **Calculate the z-scores:**

We need to convert the given average life values (793 and 817 hours) into z-scores.  The z-score tells us how many standard errors a particular sample mean is away from the population mean.

z = (x - μ) / SEM

Where:
* x is the sample mean
* μ is the population mean (802 hours)

* For x = 793 hours:
   z₁ = (793 - 802) / 19.38 ≈ -0.47

* For x = 817 hours:
   z₂ = (817 - 802) / 19.38 ≈ 0.78

3. **Find the probabilities:**

Use a z-table or calculator to find the area under the normal curve between these two z-scores.  This represents the probability that the sample mean will fall between 793 and 817 hours.

* Find the probability associated with z₂ = 0.78: P(z < 0.78) ≈ 0.7823
* Find the probability associated with z₁ = -0.47: P(z < -0.47) ≈ 0.3192

4. **Calculate the probability between the two values:**

Subtract the smaller probability from the larger probability:

P(-0.47 < z < 0.78) = P(z < 0.78) - P(z < -0.47)
P(-0.47 < z < 0.78) = 0.7823 - 0.3192 ≈ 0.4631

Therefore, the probability that a random sample of 23 bulbs will have an average life between 793 and 817 hours is approximately 0.4631 or 46.31%.