Question

You may need to use the appropeiate appendix table or technology to anower this question. A population has a mean of 300 and a standard deviation of 15 . A sample of 100 observations will be taken. What is the probability that the sample mean will be between 295 to 305 ?

Answer 1

The probability that the sample mean will be between 295 and 305 is 0.9992.

Step-by-step explanation:

To find the probability that the sample mean will be between 295 and 305, we can use the Central Limit Theorem. The Central Limit Theorem states that as the sample size increases, the sampling distribution of the sample mean approaches a normal distribution with the same mean as the population mean and a standard deviation equal to the population standard deviation divided by the square root of the sample size.

1. Calculate the standard error of the mean using the formula: standard deviation divided by the square root of the sample size.
   Standard error = 15 / √100 = 15 / 10 = 1.5
2. Calculate the z-scores for the lower and upper limits of the desired range.
   Z-score for the lower limit = (295 - 300) / 1.5 = -5/1.5 = -3.33
   Z-score for the upper limit = (305 - 300) / 1.5 = 5/1.5 = 3.33
3. Use a z-table or technology to find the area (probability) to the left of each z-score.
   Area to the left of -3.33 = 0.0004
   Area to the left of 3.33 = 0.9996
4. Subtract the smaller area from the larger area to find the probability that the sample mean will be between 295 and 305.
   Probability = 0.9996 - 0.0004 = 0.9992

Therefore, the probability that the sample mean will be between 295 and 305 is 0.9992.