Final answer:
To determine the probability that the sample mean of the amplifiers would differ from the population mean by less than 3.4 watts, we calculate the standard error and then find the z-scores corresponding to 110 +/- 3.4. We then find the area between these z-scores under the standard normal curve to get the desired probability.
Step-by-step explanation:
The student is asking about the probability that the sample mean of amplifiers will be within a certain range of the population mean, given the population variance and the sample size.
To solve this mathematical problem completely, we utilize the Central Limit Theorem, which tells us that the distribution of sample means will be normally distributed around the population mean (110 watts in this case) with a standard deviation equal to the population standard deviation divided by the square root of the sample size. The population variance is given as 100, so the population standard deviation is the square root of 100, which is 10 watts.
The standard deviation of the sample mean (also known as the standard error) is the population standard deviation divided by the square root of the sample size, which is 10 / √44. To give me 500 word answer, I would explain this concept further and provide more examples, but for the sake of brevity, let's proceed with the calculations. We want to find the probability that the sample mean is within 3.4 watts of 110, which involves calculating the z-scores for 110 + 3.4 and 110 - 3.4, and then finding the area under the normal curve between these two z-values.
Using a standard normal distribution table or a calculator with statistical functions, we can determine this probability. Typically, this will involve looking up the probability corresponding to the calculated positive z-score and subtracting the probability corresponding to the negative z-score (which is symmetrical around the mean in a normal distribution). The result is the total probability that the sample mean is within 3.4 watts of the population mean.