Random samples of size n=10 are taken from a normal population whose statndard deviation is known to be 5. Calculate the area under the sampling distribution curve for x bar between the points u-1 and u+1. That is find the probability that the sample mean lies within + or = 1 unit of the populatian mean.
To find the area under the sampling distribution curve for x-bar between the points (µ - 1) and (µ + 1), we need to use the properties of the normal distribution.
The sampling distribution of the sample mean (x̄) follows a normal distribution with a mean equal to the population mean (µ) and a standard deviation equal to the population standard deviation divided by the square root of the sample size (σ/√n).
In this case, the population standard deviation is known to be 5, and the sample size is 10. Therefore, the standard deviation of the sampling distribution is 5/√10 = 1.58 (rounded to two decimal places).
To find the probability that the sample mean lies within ±1 unit of the population mean, we need to compute the area under the curve of the sampling distribution between (µ - 1) and (µ + 1).
Since the sampling distribution follows a normal distribution, we can use a table or a calculator to find this probability.
Using a standard normal distribution table or a calculator, we can find the probability associated with the z-scores of -1 and 1.
The z-score is calculated as (x - µ) / (σ/√n), where x is the value we want to find the probability for (in this case, ±1), µ is the population mean, σ is the population standard deviation, and n is the sample size.
For z = -1, the z-score is (-1 - 0) / (1.58) = -0.63 (rounded to two decimal places).
For z = 1, the z-score is (1 - 0) / (1.58) = 0.63 (rounded to two decimal places).
Using a standard normal distribution table or a calculator, we can find that the area to the left of z = -0.63 is approximately 0.2659, and the area to the left of z = 0.63 is approximately 0.7341.
To find the area between -1 and 1, we subtract the area to the left of -1 from the area to the left of 1: 0.7341 - 0.2659 = 0.4682.
Therefore, the probability that the sample mean lies within ±1 unit of the population mean is approximately 0.4682, or 46.82% (rounded to two decimal places).